Is it possible stable coins could have their own bank since they are backed by a country's currency like the U.S Dollar which they use to maintain a stable value?
For any coin to have its own bank they don't need to be "backed" by anything. The criteria is
to be centralized which stablecoins already are. That means the answer is yes they can have their own bank.
After all the dollar itself is not backed by anything and it has its own banks simply because it is centralized!
This is exactly why there is a flaw in this design. You see centralization and blockchain technology don't mix well. It is like addition of contradictions!
By the term "Own Bank" I think it means a store of value but isn't all currency and assets have this already? Banks are known to be centralized but I think a decentralized bank is also possible. Dunno bout you but I think dollar is backed by something and anything that are legit is also like it. The backer of dollar can be literally banks and maybe the governments too?
Nothing is perfect in this world but each must have their own flaws. We only need to accept this reality or we can still go for the other if we don't like what the other is offering. I am sure we are not forced to use them. Blockchain itself must offer transparency and so as centralization. They do mix well actually.