The main use cases you may be interested in are to 1) make it so that users connecting to you get a faster experience - since they can connect to a geographically nearby cloudflare server, rather than your distant server and 2) make it so that cloudflare can absorb large denial-of-service attacks that your server couldn't otherwise cope with.
So why couldn't Cloudflare set up what I'm suggesting? Are there technical reasons, or just lack of demand?
But it sounds like you're not very familiar with the importance of the private key. If anyone else other than the bank obtains the bank's private key, the bank would consider that a serious failure, since it means others could impersonate them. The whole point is that you shouldn't give your private key to anyone else, and that without that key, others can't impersonate you.
This "keyless SSL" scheme allows the bank to set up an entity it controls which knows the key. This entity delegates to cloudflare the ability to pretend to be the bank on a request-by-request basis, without divulging the key to anyone. If cloudflare gets compromised, the bank can stop that delegation on demand, the compromise is closed and the key is kept safe.
The only way for them to set that up would be to become a CA, get browsers to accept their master certificate and then issue their own certs for their clients.