Is the only way to access streaming channels such as Youtubetv or Criterion Channel to a third generation appletv box through AirPlay?

No problem

As I said that model only has those preset server 'channels'. So providers who already had a server version for that model have steadily been dropping support. The latest was HBO, and when that happens the icon for that service will disappear.

Correct. The 3rd gen is not able to run apps and only offers preset channels via server. New services are opting to create apps, and thus not available, and existing providers have also been dropping support for their server versions.

You're welcome