[RPG] Does a permanent enhancement bonus to Intelligence increase 1st-level spells known

In the end, you're asking for an opinion since they borked the RAW and haven't errata or FAQed it yet. I would rule it as an alchemical bonus for exactly the reasons you already list - it seems like it's the clear intent. It's a low level extra mutagen so it's only a modest boost.

• If a level 1 alchemist has Int 11, he can create 1 1st-level Extract Per Day.
• If a level 1 alchemist has Int 12-19, he can create 2 1st-level Extracts Per Day.
• If a level 1 alchemist has Int 20-29, he can create 3 1st-level Extracts Per Day.

But, barring other influences (e.g. archetypes, feats, magic items, special abilities, traits), that's all he can create. He can't create 2nd-level or higher extracts, and he can't use the higher-level bonus Extracts Per Day to create lower-level extracts.

Although a high Intelligence score lets an alchemist create more higher-level Extracts Per Day, an alchemist can't create additional higher-level Extracts Per Day those until he has a number instead of a dash of Extracts Per Day of that extract level according to Table: Alchemist.

So while a high Intelligence remains extremely valuable for an alchemist, that high Intelligence doesn't let him create Extracts Per Day of an extract level beyond what his alchemist class level dictates he can create.

The alchemist's Extracts Per Day follow the same rules as Bonus Spells Per Day for spellcasters.

What would happen if an alchemist could

I'm mainly guessing, but I think you were thinking that because of an Intelligence score of 18, a 1st-level alchemist would be able to create 1 additional 1st-level, 2nd-level, 3rd-level, and 4th-level Extracts Per Day if he could gain access to those formulae. The game expects, however, according to Table: Alchemist, a 10th-level alchemist to be able to create 1 4th-level Extract Per Day. In other words, an ability most characters would wait to use until halfway through their careers (i.e. creating 4th-level extracts) would be available to a 1st-level alchemist with Intelligence 18.

The game doesn't expect a level 1 alchemist to use 4th-level extracts like freedom of movement and greater invisibility even once per day, and you'd likely find such a character overcoming challenges far beyond his expected abilities, leading to DM frustration at the DM's inability to challenge such a character without killing him outright and, perhaps, boredom (or terror!) while playing a character who can casually overcome challenges far beyond his ability in an extremely limited manner. You can play that way, but you should be aware you're choosing to change the rules, and your experiences with the game will be vastly different than those who are otherwise playing the same game.

A minor rules quirk

The supernatural ability alchemy says, "An alchemist begins play with two 1st level formulae of his choice, plus a number of additional forumlae equal to his Intelligence modifier." That last part means that an alchemist always begins play with 2 1st-level formulae but that his Intelligence modifier lets him select additional formulae of any level. He can't use those higher-than-1st-level formulae when he's a level 1 alchemist, but he could totally have in his formula book as many 2nd-level-through-6th-level formulae as his Intelligence modifier allows.

Again, the level 1 alchemist can't yet create extracts with such formulae, but he could—if he encounters appropriate folks during the campaign—trade that formula knowledge for other goods or services from a higher-level alchemist that desires such formula knowledge. Of course, such a higher-level alchemist may, instead, opt to simply disappear the low-level alchemist and abscond with his formula book, but a low-level alchemist choosing to have high-level formulae in his formula book is an option, albeit an unusual and potentially dangerous one.