I have always believed there is room for an $xx/year service in the US that provides a virtual bank account number so you can easily switch banks in the background. The decrease in switching friction may actually make banking interesting/competitive again.
Fidelity and other major online brokerages do this with their money market accounts. It doesn't chase an optimal yield like what you are proposing, but that would be interesting. Instead, they use a large basket of money market and savings accounts at US banks and interest payments on your savings are drawn from a random bank in the basket each month. Because the federal reserve sets interest rates, it's difficult to chase yield in bank savings accounts. You'll always do better with bonds and stock dividends.
Once people started using any such service, they would realise that their customers are locked in / can't switch easily, so they'd raise their prices to $xx+1 each year... and then you're back to where you started from.
There's no need. No one knows your savings account number, just your checking account number.