This is a combination of history and what appears to be a small but significant oversight. The advantage of the Haversack over the Bag has always been that the Haversack always has what you're looking for on top, as compared to the Bag which is a disordered bag of stuff that's harder to sift through the more it holds (and it can hold so much).
The designers appear to have retained this distinction in the items' described function, and then made it immaterial during development by copy-pasting the standard 5e item-interaction boilerplate into their descriptions without making any adjustments to implement the functional difference they kept in the descriptions. (Easy enough to do, since when you're doing the rules polishing on the Bag you're not thinking about the Haversack, and it all looks fine, right?)
Traditionally (I mean back in AD&D, since in 3.x this is likely the kind of thing DMs would just handwave away), digging through a Bag for an item wasn't feasible during combat, so the discovery of a Haversack was a significant upgrade in, well, handiness. The capacity difference meant neither was strictly superior, but each had different pros and cons (the marginal utility of being usable in combat making the Haversack especially desirable, but still not strictly superior in every way). But none of this was nailed down in rigid action-economy terms then — there was no such thing as strict action economy terminology, it was just how the items' descriptions said they worked. The rarity difference in the 5e items appears to reflect a design intent to limit access to the especially-desirable item usable at combat speeds; later negated by failing to implement this small but significant distinction during the development stage.
So it just seems to be an oversight. RAW, the Haversack is the clear loser. For many DMs, the question ends there (plus a bit of head-scratching at the designers). However, 5e DMs aren't bound by RAW and are encouraged to make 'fluff' matter in their games, if they so desire.
If you want to emulate the traditional utility of the Haversack, make Bags of Holding able to be interacted with using an action... but it takes multiple before the user finds what they're looking for. You can either nail this down (at which point you're into making house rules to taste to determine exact number of actions), or just say that it's only possible during non-combat time and handwave the exact time it takes. No matter how you implement it exactly, this will leave the Haversack as the true champion of handiness that's usable with a single action to get exactly what you want.
Arguably, this isn't your problem. In most circumstances, you don't give loot to specific characters - you give it to the party, and let them sort out how it gets distributed.
Yes, this could theoretically lead to one player hogging all the magic items, but that's a solved problem; As long as your players are aware that they can negotiate how stuff gets shared around, they'll generally spot unfair situations, renegotiate as necessary, and generally agree on equitable solutions on their own.
If one player's lack of magic items is dragging the party down, that's the whole party's problem, and most players will realise that pretty quickly; Even the most selfish of players will agree it's a good idea to share the wealth if it improves their own survivability.
If magic items have already made one player much more powerful than the others - Well, it's not like magic items are bound to the first person who picks them up; Your players can always just redistribute them where they'll be more useful.
If your players don't find an uneven distribution of loot to be inequitable - Well, then there's no problem to solve; Everyone's already having fun, and that's what the game is all about.
So, to repeat myself, this isn't your problem: Working out how to distribute loot is just another part of the game. Heck, some players find it one of the most fun parts of the game: Working out how to effectively use available resources is core gameplay, and deciding how magic items should be distributed for best effect is part of that.
Best Answer
Rarity and usefulness/power are very weakly correlated, in my experience.
This has been discussed elsewhere (reddit, GitP, ENWorld), primarily when 5e was first released. I'll point you to my favorite resource, the Sane Magical Prices Index by GitP user Saidoro: I've used it for years in order to gate items in campaigns and have been very happy.
First Saidoro establishes that item rarities are obviously bunk in plenty of cases: compare (for yourself) the broom of flying/winged boots to the wings of flying or boots of levitation. We see there functionally-equivalent or even weaker items "rated" two tiers above comparands.
After dividing the items into comparable classes (consumables, combat items, utility items) Saidoro and other posters spent months discussing/debating the utility/power of each and set a scale, in gp, for almost every magical item in the DMG. Follow the threads both on GitP and on reddit/ENworld (linked in Saidoro's .pdf) if you'd like to see more of the reasoning that goes into each ranking/valuation.
The long and the short of it is that while one might argue with some of the valuations, there's no question that this list, compiled by many actual players and much more finely-graduated than the rarity tiers, is a better guide to power than is rarity.
So I threw all the prices and rarities into my statistics software and ran a regression with rarity as the explanatory variable: the R2 value is ~0.0236. That tells us roughly 2% of the power/utility of each item--as judged by users and tabulated by Saidoro--is attributable to the item's rarity.