Problems with using FlashPlayer with Mozilla Firefox

My guess is that this is related to DRM.  I'm not aware of a similar widespread problem affecting Firefox users and DRM content in a generic way.  It's more likely that there's something stale that's been cached.  The ecosystem for authenticating and validating your access to rights-managed content is very complex, involving a number of third-parties, and there are lots of possible failure points.

I'm guessing that reinstalling Flash Player is just getting you a new DRM credential that's valid only for the session, which is odd, but explains why it works.  My first recommendation would be to clear your browser cache, and blow away all of Flash Player's stored data.  Alternatively you could launch a private browsing tab, and see if the problem works there.  Either of these should get you an instance that doesn't use any existing stored credentials.

Go to Control Panel > Flash Player > Advanced > Delete All under Browser Data and settings, and also click Deauthorize This Computer.  This should blow away any stale credentials and DRM licenses that have been issued.

At that point, hopefully the problem is solved, and it was just a stubborn cached file.  If not, something more nuanced is going on.

Do you happen to be running any ad-blockers or anti-tracking plugins that might prevent JavaScript in this authorization ecosystem from working as written?  That might explain why your credentials work on the first try only...