This would be wonderful. It would give younger people a chance to play the old generations and receive a reward in their latest version for it, as they could get those lovely Pokemon. As for Nintendo and Game Freak, letting people take in so many Pokemon from so many versions, some of which would be useful (at least as an addition to the Pokedex) and some of which would just have sentimental value, would certainly be a nice way to get people to pay for the Bank.
In regards to what games to include, I would say having Red, Blue and Yellow would be great. For Japan, it would be Red, Green and Yellow, so maybe adding in the old Blue for them would work out, then doing the same for the U.S. and naming it Green (with an explanation on the shop as to what it is. After Gen 1, the Generations are pretty easy for what to include. I'd skip the remakes, personally. Maybe something can be done so that our Pokewalkers are useful? I didn't put much thought into that though.
I wouldn't mind seeing our dear friends in spin-off series too. Specifically Coliseum, XD, and the Stadium games. Those could go on the Wii U and transfer into the Bank maybe? I don't know how possible that is.