This was intentionally changed after 5 years when it was originally an express behavior in code.

https://bugzilla.mozilla.org/show_bug.cgi?id=1968048

There was a plea to keep it working two months ago when the change was made, and a couple of us commenting on a secondary bug report that listed this as a regression because who would expect that to suddenly break without notice.

https://bugzilla.mozilla.org/show_bug.cgi?id=1973769

I'm sure this issue will be shunted to the Mozilla Connect site and treated like it's an entirely new request, but for what it's worth, they added the address-bar-share-button just recently, got a bunch of complaints about it, never responded there, and now reverted it in 141.

I'm posting this here in part because it's not obvious that there are several distinct places to look to figure out what's going on with changes like this.

I think the workaround for now is to put a "Private Firefox" shortcut on your home screen. That opens a new private tab. See the "Add a private browsing shortcut" section of this article: Private Browsing on Firefox for Android.