You mention that you just got the ATV. Were you using Zwift previously on a different device? If so, make sure you are logged out of that device as this sounds like the behavior of being logged in on multiple devices at the same time.
Shuji at Zwift HQ here. I’m looking at your server logs for today Jan 4. The thing that catches my attention is that there’s one login event from the Apple TV, but a series of subsequent unmatched log out events. Each log in should have a matching log out. I’m not sure how that’s happening.
Even without this oddity - you must ride at least 2.6 km / run 0.8 km in Zwift for a session to save. Some other reasons why a session may not save are covered in this support hub article