Why counter is the shared secret? In TOTP time is the counter and it is obviously not secret, so there is no reason to think the counter would be secret as well. Clients can sync their counter to match the server.
In HOTP, the secret counter is not independent and must remain synchronized between client and server.
A counter that can be synchronized on demand is kinda superfluous --- not really secret and not terribly relevant either. All else being equal, an attacker can sync up just as easily as a legitimate client so why bother with the counter?
I expect HOTP exists somewhere out there in the real world but I have yet to encounter it. Every 2FA I have actual experience with has been TOTP.
A unique counter for each authorization attempt ensures the resulting key is different for each attempt, which makes replay attacks not possible. I agree if you sync the counter two ways, it is better to use a "nonce", a totally random secret each time.