Version 2.3 is now out and it works well for this use. (I use it myself now on my live site.)
Let's assume you schedule two YouTube Live events. They both must be set to Public.
Before you're live, the plugin will display the YouTube Live countdown to your next event.
When you go live, the countdown automatically changes to your live video.
When your live video is finished, the player stops. After a page refresh, the countdown to your next live video will display.
After your next live event has ended, and if you don't have any other scheduled events, the player goes away and the "not live" message displays, or you should be able to leave this blank to let your last YouTube upload display.
Update to 2.3 and try it out!