You can obviously play any way you want, but the point of a bank alt isn't that of a gear-holding mule, it's a transfer point.

1) send your mail there, bounce it back. Suddenly every mailbox is now your bank, in effect (for non soulbound stuff). If you're a crafter, no need to fill your bags with raw mats - just bounce it off your bank alt and then they're available anywhere there's a mailbox.
2) Again, play your own way but NOT taking advantage of the AH is deliberately handicapping yourself. Having 3 80's all with fast flying, I don't really do dailies but each has well over 4k gold and my single bank alt can easily pick up 1k g per day on a good AH turn.
3) I use my bank alt's bank slots (and guildbank, since he was able to buy a guild off someone for 10g) to accumulate 20+ stacks each of minerals, gems, and primals so I have no problem regemming as needed, or levelling up a toon's jewelcrafting/blacksmithing etc. I was accumulating lowbie herbs for a while, and was able to clear the bank mostly when Inscription came out....probably a cool 2-3kg with essentially no effort.

And frankly, if you're only filling your bags after a DAY of questing, you're doing it wrong. At pretty much any level typical questing activity, my bags fill in about 20-30 minutes, tops. Skinning alt sometimes half that. In some zones with a bunch of gather quests, and if you have bad luck in acquiring bags, you can have 50% of your available slots filled with quest crap.

Really, I can understand RP'ers not wanting to do it, but I can't comprehend really NOT wanting to use a Bank alt. For me it would be tantamount to vowing only to use grey items or some other deliberate self-crippling.

FWIW frankly I think Blizz's "insta mail within the account" is a n00b crutch. I'd prefer the 1 hour transfer for gold or items, to cripple the goldsellers and force SOME need to 'manage' your inventory to *some* degree.