I hate to break it to you, but throughout the history of the Obama Administration, any time any person from top to bottom has stepped down from their position, without exception that person has been replaced by someone even less competent. So when Frieden resigns, we will get someone even worse in his place.
The public and media may be forced to pay closer attention to this one though
I reject the “job insurance” threat (arranged so successor will be worse, so job occupant won’t get canned in fear of greater idiocy). If we end up with someone worse, keep removing incompetence until someone sensible shows up to actually do the job. For example, removing Obama would get us Biden as POTUS - ok by me.
Don’t let the threat become viable as a tactic. Solve problems until you have a solution.