Thanks. Yes, this is something what I think of. Does it show on the screen what operation you confirm with U2F? Or just "U2F authentication"? But cord makes it far less convenient unfortunately.
I use it for gpg, ssh(gpg-agent) and u2f. These are official applications that you can install on the device that does the above. It doesn't show the operation, just the website trying to access.
I'm not sure if it supports U2F. If it does, I haven't used it. It just seems to prove that what you were conceptually describing can exist, and at a not-completely-unreasonable price point.