A few different reasons from what I’ve gathered lurking around various communities. Full disclosure: I’m not in the aviation industry, just someone with an interest.

• Controllers are extremely professional almost to a fault. If they walk that puts more strain on an already strained system which endangers the public
• Related to this, the public is still generally on their side or at least sympathetic to the situation. If they strike and there’s an incident, they will be blamed. the admin would love for this to happen. They’re already trying to turn public sentiment against them (see comments by the transportation secretary about starting salaries which are demonstrably false).
• Their current union is not in favor of any work action
• They are already being threatened with firing if they strike under the guise of sick leave
• For most of them, this isn’t their first shutdown
• The controllers that’ll feel the pinch first are those working at smaller, less busy facilities. One or more of these shutting down completely won’t impact much.

Again this is just my observation. Anyone with more insider knowledge feel free to correct me.

The orange turd already promised people would not get any back pays.

Well no he didn't. He stated that about furloughed employees. The ones not going to work. The ones working are entitled to backpay paycheck.

Additionally, and as always, he's talking about things he doesn't really understand. Federal backpay for furloughed workers was codified into law. Meaning they'd have to pass a budget with a line specifying that furloughed federal works won't get paid and nullifying the previous law. That would be quite a bad move for most congressional representatives. Almost a garuntee to lose a lot of votes.

I believe they're legally forbidden from striking and have to be careful it isn't seen as such.

Oh say is that plane really

falling from the sky?

O'er the land of the free

Fuckin' poke in the eye