I just realized that the version I have (23306.3308.2524.1744) is a preliminary version.
How do I get that preliminary version? I otherwise have to use the classic edition until then. Mine currently shows 23285.3607.2525.937.
Try this - https://statics.teams.cdn.office.net/production-windows-x64/23306.3308.2524.1744/MSTeams-x64.msix from a post I found here - https://answers.microsoft.com/en-us/msteams/forum/all/register-the-new-teams-as-the-chat-app-for/d67da9c0-2904-4bfe-b0e9-d5e8a35b5d07?page=2
After installing this I went to Settings -> About Teams and this prompted me to downgrade my Teams version back to 23285.3607.2525.937.
When I did that my status magically appeared :)
Hope it works for you!
I am now at 23306.3308.2524.1744. I've already uninstalled the prior ones before installing this one, and even restarting it, but it didn't ask me to downgrade. And, it's still not showing any status within the icon on the system tray. Darn! The only reason why I didn't have the 23306 was because I had downloaded the x32 version instead of the x64 version. 23306 is only available on the x64 version, apparently. And that x64 also fixed the "Register the new Teams as the chat app..." Thanks! I just wish the status one works, but may require downgrading, which mine doesn't do.