Some weapons have to be sheated before being put into a handy haversack or they risk perforating it. You still need to unsheat them after drawing them, unless you have the Quick Draw feat or the Magic Item Compendium weapon crystal that doubles the feat.
Using a club or some other weapon that does not need to be sheated could be a nice alternative.
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.
Best Answer
Pathfinder explicitly calls out the rules on extradimensional spaces, noting that
The d20srd doesn't appear to have such a rule (except for the exception about horrible things if a bag of holding and portable hole interact with each other), so it's fuzzier.
That said, keeping a haversack in a haversack (or replacing either with a bag of holding) should be fine - the contents of the inner container aren't available until removed from the outer container, but nothing bad happens.
In either case, however, both the 3.5 and PF description of the Handy Haversack say
Since the description calls out "the wearer", I don't think you could pull an item from the inner haversack out with two move actions: after removing the inner haversack from the outer, you would be holding the inner haversack, not wearing it. Strictly, to take advantage of the move action to withdraw an item, you'd need to spend some time putting the inner haversack on (and, presumably, taking the outer one off, though their being slotless makes that odd).
Do note that Pathfinder added the Secure Paypack, which is a slight upgrade to the Haversack (it has two additional one-cubic-foot or 10-pound compartments, and some nifty fluff).
Also, note that Bags of Holding are less expensive per volume/weight than a Haversack. For things like "a cooking kit, animal feed, a climbing kit, and a tent", toss them in the BoH and don't worry about the extra time it takes to get the thing out (none of those are meaningfully usable in combat aside from the climbing kit, which is only useful if already donned).
That said, this GM frankly doesn't care that much about how much is in your haversack (nor do most that I've played with), especially if it's just being used for wands (though, note the Efficient Quiver; wands are basically arrow-sized, right?), potions, scrolls, cash, gems, and the like. In a campaign where I did care, I'd likely be extremely receptive to players simply buying a better Haversack if they cared that much about having instant access to that much stuff. ... then again, this GM is currently running a monk with a Secure Paypack, a Type 2 Bag of Holding, and a Portable Hole (carefully stored in not-the-paypack-or-BoH), so...