Banks were necessary in Daggerfall, because coin had actual weight. If you wanted to buy an expensive house, you had to load up a cart with gold coin so you could haul it over for purchase. Or you could go to the bank for a letter of credit. It wasn't really feasible to haul around a million gold coins. That's where banks came into play.
As someone who enjoys the concept of way too much realism, I would welcome banks, and weighted coin, and letters of credit for that matter.
In Daggerfall gold weight was 0.1 the same as light plant ingredient. so 1000 gold weight 100. Now add that prices in Daggerfall was far higher than in Oblivion and even Morrowind. You could buy daeric armor in shops but it easy cost millions, but at high levels you could afford it.
As i remember elven items had higher value/ weight than gold so you had the nice drop gold button you used in dungeons if you was stupid enough to pick up gold at medium to high level.
In real life you can carry around 10.000$ without it weighting you down. If money had weight you should have copper, silver and gold money where one copper would be the lowest value, typically a potato. a gold coin would be worth 100 copper or more.
Yes you might end up having to drop copper but a gold coin would typically be 100 times more expensive. You would still want some arrangement like letter of credit buying a large house or similar. Banks also offer other options like interest and investment.