Isn't that what Mom, at least in Gen 2, and the PC were for?

I never really cared for storing items, since we can carry an unlimited number as of Gen 2.

And since the amount of money lost versus gained from battles is negligible, a bank seems unreasonable, and pointless.

Now if the amount of money gained was harder, and the amount lost per loss and/or purchase was much more punishing, then the bank would be much more useful. Especially if it gave interest for storing the money there.

At current, there's no reason and almost no punishment for carrying all of your money and items on you, thus leaving the bank an empty, pointless building.