Strikes cost striking employees money (because they aren’t being paid while striking). In this case, however, they aren’t being paid even when not striking.
Strikes are a drastic measure. Well, hello! Isn’t not being paid so much as a cent for twenty-five days and running a drastic measure?
Figure out which agencies would impact Trump voters the most, and shut those down with strikes.
But really, why should Trump think he can: shut down the government (and he’s made it clear in his comments that it’s his shutdown), get away with not paying people, and expect them to continue working as if nothing has happened?