In TOTP, time exists independent of both client and server and as you point out, is no secret to either.
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.
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.