If every bank produced their own stablecoin, there would be a huge problem of interoperability. For example, I have Bank of America coins but the merchant only accepts Wells Fargo coins. A few coins would be manageable, but if there are 1000 coins it would never work.
So, that brings up the next problem. If there can only be a few coins, it would put small banks out of business because they couldn't compete with the big banks.
Regardless, the idea of a CBDC really doesn't make sense. A CBDC would not provide any benefit beyond what banks and central banks already provide. At least in the U.S., the Federal Reserve is run by banks, so it would not do anything that would put banks out of business.
What's the point of using a stablecoin that is backed by fiat money, when you could use fiat money directly? I believe that stablecoins have no other utility besides crypto trading.
The advantage of the stablecoin is that you can send the coins directly, something that you would not be able to do with a CBDC. In order to allow sending coins directly, the central bank would have to give up control of the money, and that's something it will never do.
You can't use fiat money online
What are you talking about? I use online dollars to pay my bills and send money to friends all the time. I use credit and debit cards to buy stuff with online dollars.