Arcanist, Sorcerer, and Wizard are three of the most dominatingly-powerful classes in the game

Each of these classes, built right, can do just about anything, and in many cases can do very close to everything. The spell list they have access to is the best in the game, and spellcasting is the best class feature in the game.

These three can nullify a huge array of problems with a standard action. They are all extremely capable of getting new solutions in those cases they run into a problem they cannot already solve.

At very-low levels, these classes have somewhat-low stamina, and their spells are not nearly as dominant as they quickly become, but smart and careful play can ensure statistical advantages in almost every endeavor. As they level up, their stamina and power increases exponentially, and it is not long before they are clearly superior to almost everything else. Around 7th-level, at the latest.

This does, however, rely on careful and creative play. Extremely thorough knowledge of the massive sorcerer/wizard spell list, and what spells are best for which situations, is required. It takes a lot of work and mastery to accomplish. Just as the ceiling for these classes is very high, rock bottom is a very dark and dank bottom indeed. That said, they have exceptional ability to fix any mistakes they make; even if they play very poorly for a long time, they can very suddenly become powerhouses by simply preparing better spells (the sorcerer is, obviously, somewhat worse off in this regard).

Summoner and witch are close

The summoner and witch are very close in power to the three listed above. The eidolon is an incredibly versatile and powerful class feature, summon monster offers a great deal of utility, and the witch is pretty much a lesser wizard.

The summoner has the side-benefit of being much easier; it’s pretty easy (and effective) to hide behind the eidolon. The witch is actually kind of harder, though probably more powerful; the familiar is a massive, glaring vulnerability, and the spell list just isn’t quite what the sor/wiz one is. But then, the witch gets to heal status ailments, which none of these classes get, and though it’s supremely limited, that could be a big deal (see below).

Alchemist, bard, and magus are more well-rounded

These classes are not nearly as powerful as the above, but like the summoner they can be a bit easier, particularly the magus, as they are sturdier before spells come into play. In particular, they also allow for more up-front characters, which is not only something many enjoy, but also is often a useful platform for the more magely casters to put buffs onto.

These classes are reasonably competent, and are in a pretty sweet spot, power-wise. They also add a bit of variety compared to the above (though the summoner’s eidolon does a depressingly good job filling the melee role). A group at this tier, or at this tier with support from the above, will be very competent.

The alchemist is also very important because it is the only class here that gets lesser restoration and restoration, which are otherwise major gaps in the capabilities of arcane spellcasting.

I have no experience with bloodrager, investigator, or skald

Just to be upfront about it: I have no practical experience with these classes, unlike the above. So take the below with a grain of salt.

That said, the bloodrager’s spellcasting is extremely weak, and nerfed rage powers are not a particularly exciting thing to get in return. This class looks like it could be quite fun in a low-power game, but I would not recommend one in a game alongside the above.

Skald looks better; its spellcasting is closer to the bard’s, and it gets enough barbarian powers to be interesting. I actually want to see this class in play at some point. Probably plays well at around the bard tier?

The investigator seems to be on a similar page as the skald, though the existence of the vivisectionist alchemist makes me wonder what the point of it is. I rather like the concept, anyway.

One major thing arcana is lacking: healing things other than HP

HP healing won’t really be a problem; even the bard can handle that well enough with a wand of cure light wounds (though make sure the party does get such a wand somehow), and the witch is actually pretty good at it, but healing status effects, ability damage, and conditions is going to be much harder. The cleric list is pretty much the only places you find really good healing of that sort.

The alchemist and witch are the stars here: the alchemist gets the crucial lesser restoration and restoration spells (though he needs the Infuse Extract discovery to share them), and the witch gets the various remove ailment spells, plus heal at very-high levels. Between them, they cover the major healing options that are otherwise missing from arcane spellcasting. The cleric still has major advantages (not least of which being that the cleric can handle both of these things), but if both of these classes are represented in the party, they can probably handle these things about as well.

If either or both of these classes does not get chosen by players, be aware that anything that inflicts an ongoing status effect is likely to force the party to retreat, to find a temple to get it dealt with. That, or if absolutely forced, to throw themselves into later encounters at a massive disadvantage. Deciding how you want the campaign to play out with this reality would be really important. Items and institutional resources are likely going to be necessary, which means you, as DM, have to make them available.

There are a number of reasons why the character's book knowledge may not be useful at a given time. There are also a number of reasons why the character's knowledge wouldn't be as extensive as your player seems to think.

1) Book knowledge isn't practical experience.

You can read all the books in the world about riding a bicycle, and you'll still fall off the first time you actually try to do it. There's tons of sensory information that simply can't be reduced to words and absorbed, to say nothing of muscle memory.

2) Fantasy settings are not usually full of technical manuals.

There are no printing presses; everything is written by hand. Nobody writes a book explaining the basics of sailing, because that would be a massive waste of effort; it would be far easier to teach people how to sail by actually taking them sailing. This is especially true for things that are basic day-to-day skills for a decent segment of the population. Why would you write a book about how to do X when everyone you know already knows how? Books that include the application of these skills will usually be journals, and will either gloss over the details so no knowledge is necessary, or simply use sufficient explanation for someone who already knows how to understand what is being described.

3) Information in books may be highly localized to a specific place and/or time.

Knowing exactly how kayaks are built & operated by the jungle tribes in the distant land of Gobbeldey-gook doesn't help you operate a Gibberese catamaran. Knowing how ancient Gibberese oared catamarans were built & operated doesn't help you operate a modern Gibberese triple-masted caravel. Any books a centuries-old elf has read are likely to be tens, hundreds, or even thousands of years old, and has a higher chance of being out of date than the elf might realize.

4) Information in books may be straight up wrong.

Unless the book was written or dictated by a major deity whose portfolio includes honesty, its source is capable of deliberately lying, being factually incorrect, or both. History books in particular have a high risk of being revisionist, potentially to the point of being a complete whitewash. The idea that historians should faithfully record what actually happened, regardless of who it makes look bad, is pretty much limited to modern liberal democracies; in a fantasy setting writing a history book that doesn't flatter everyone powerful in the area is a good way to get a visit from some soldiers, assassins, or "adventurers".

5) Not every book is in Elven.

There are a wide variety of languages in the Forgotten Realms, and plenty of books won't be available in languages the character speaks (remember, translating a book involves writing out the translation by hand). If you can only read Elven, then most of the knowledge you get from books will be on topics that an elf thought was worth writing about.

6) Literacy might be rare.

In real world history, most people in the medieval period D&D is based off of didn't know how to read & write. This not only substantially reduces the number of people out there writing books (why learn to do anything that pays less than being a scribe if you already know how to read & write?), it also substantially limits who writes books. If only nobles and priests have both the knowledge and the spare time to write books, you should expect the list of topics available to be fairly limited. Admittedly, literacy may not be as rare in D&D settings (PCs can always read & write common, for example, though PCs are by definition a special case), but it's almost certainly not the >90% we're used to in the modern day.

7) Does the character's backstory actually give them time to read that many books?

Part of being a wizard is doing a lot of reading & studying, yes, but it's specifically studying magic. Wizards don't have innate access to magic the way sorcerers do, or clerics, or warlocks, or just about any other class; wizards get magic by studying and practicing magic a lot. Becoming a level 1 wizard is a lot like getting a PhD in neurochemistry; being brilliant helps, but it still takes years of focused effort. Even being an elf doesn't give you enough time learn magic and read up to expert-level on sailing, wilderness survival, monster lair layout, and all the other million and one things it can be helpful to know when you're an adventurer.

8) What world-famous library were all these books in?

When books are hand-written, copies are hard to come by. Many books will only ever have one copy in existence. A library with copies of even a quarter of the material plane's extant books would be famous across the planes, and almost certainly wouldn't allow random elves to wander in and spend a century or two perusing the stacks.

TLDR: Knowing lore, especially about old stuff, is pretty reasonable (though an appropriate knowledge check should still be required). Knowing how to do things should generally be a hard sell; few people have both the practical knowledge necessary to produce a book about mundane skills and the spare time & literacy needed to actually write a book (or hire a scribe to take dictation, I suppose).

Non-Canon Alternatives

The books don't really explain what elves actually do with all that extra time that doesn't involve being trained in every skill ever as well as being an accomplished mage, archer, and poet. Here are some options that rest on unofficial attempts to explain why elves are mechanically at about the same competency level as humans despite having been an adult for decades or even centuries longer.

The elf may not really remember everything he read while in long-time.

There's an excellent thread on the GitP forums, So You Want to Play an Elf, which, while written for 3.5, is largely fluff-focused and thus fairly easily translated to 5th edition. To summarize the relevant bit, it argues that elves perceive time differently than other races do: what shorter-lived races consider normal is short-time to an elf, and they only perceive the world that way in stressful situations that require them to do something out of the ordinary. Elves (unless crazy or very unlucky) pass the vast majority of their lives in long-time, which is a lot like being on a very relaxing sort of autopilot.

If you go with this interpretation of elves, then the elf probably has trouble consciously remembering things he read in long-time. Alternately he may be one of those lunatics who avoids or is incapable of entering long-time, but that would almost certainly give him a pretty poor reputation in elven society (which might very well limit his access to an endless supply of books).

Elves may focus on learning largely useless meta-knowledge, rather than picking up a broader knowledge base.

Another possible explanation for why elves aren't masters of every possible skill or talent is that when an elf learns something, they obsessively learn everything about it, spending a lot of time picking up trivia that few shorter-lived races would bother with.

A human trained in Arcana could probably tell you that the evil wizard is casting Hold Person. An elf trained in Arcana spent an extra thirty years picking up the knowledge needed to tell you that not only is the evil wizard casting Hold Person, the specific version of the hand gestures he's using is typical of the Broken Mirror school, founded by the tiefling Uk-shae 2217 years ago. The elf could further explain the history of that particular school/style of arcane magic, as well as each of the seventeen known variants of Hold Person's somatic component and their origins, plus which variant is a quarter of a second faster, which variant is a full 6% more mana-efficient during the current phase of the moon, and then follow up with a discussion about the philosophies of the various scholars (probably elves) who did the research to determine said speed and mana-efficiency.

With this fluff/crunch interpretation, elves definitely know tons more than shorter-lived races, it's just that most of that knowledge is completely useless minutiae. Elven bards could be the exception to this rule, or they might simply know a truly horrifying amount of trivia to match the breadth of their knowledge/skills.