WatchOS public beta,

No, it is believed the reason there is no watchOS public beta is because there is no way to downgrade back to the current release version of watchOS, which can still be performed with macOS, iOS, and tvOS, through a USB connection.

You are probably right but when Apple open the public beta for iOS 11, he also open public beta for Watch OS (this was the first time, this doesn't happen for the previous beta). So either it is a mistake, or Apple plan to release a public beta but don't do it yet.

On the other hand, the profile for Watch OS is called "iOS Beta Software Profile" as for the iPhone and not "WatchOS Beta Software Profile". Weird.

Yes I also have the Betas Profile installed, and after installing a few weeks ago it updated after installing the profile, but nothing since then. Under about my watch has version 4.3 (15T212) is that the latest.