Open Xbox Companion App (not the Xbox app, though that may also work)
Sign in to the account you're using in the launcher in the Xbox Companion App. You do not need to do anything minecraft-related in the app, just sign in to the app itself directly.
Relaunch the launcher
It needs to be the Xbox Console Companion app specifically. Even though I was signed into the more modern Xbox app, I was not signed into the older XCC app. Once I signed in and opened the launcher back up, it showed as being online and Play Demo was replaced by Play. I hope this helps someone else.
I am also having this issue. I've attached my launcher_log.txt file, and a screenshot of the launcher login screen - my microsoft account name and profile icon are missing/broken. Removing and adding the account does not help.
Dragoshi
I would just like to update people here - I've found a solution that's helped me while looking at other open issues: https://bugs.mojang.com/browse/MCL-11510?focusedCommentId=1170531&page=com.atlassian.jira.plugin.system.issuetabpanels:comment-tabpanel#comment-1170531
Close the launcher
Open Xbox Companion App (not the Xbox app, though that may also work)
Sign in to the account you're using in the launcher in the Xbox Companion App. You do not need to do anything minecraft-related in the app, just sign in to the app itself directly.
Relaunch the launcher
It needs to be the Xbox Console Companion app specifically. Even though I was signed into the more modern Xbox app, I was not signed into the older XCC app. Once I signed in and opened the launcher back up, it showed as being online and Play Demo was replaced by Play. I hope this helps someone else.