It hasn't happened in America, but I think people severely underestimate how devastating a real strike can be. Remember that the ports strike in October only lasted 2 days but estimated costs were already in the 10 (or even 11) figures.
A full on government walkout for a day would fix a ton. They won't care, but even their voter base wouldn't ignore the late payments, cancelled appointments, and overall confusion a day would do.
Every society is X missed meals away from anarchy. For a society like America that hasn't experienced famine en masse in almost a century, that X wouldn't need to be too high. They can ignore their kids but not their bare bones basics.
I'm all for the concept of a general strike, but I think the general public would be very unsympathetic with a government employee strike. People would be racing to put their name as someone willing to replace a striking government worker.
I don’t think you realize how hard it is to fill these critical jobs. You would need skilled labor at pay substantially below the private market rate. A lot of these talented people work for the government because they’re patriotic. And this admin is doing its best to piss off these people.