its a mind game. a lot of union collective bargain agreements have no strike clauses. that's the primary reason employers have to come to the table. almost all the "rules" have all these off ramps and cool off periods to stop workers from stopping their work, because its too damaging to capital formations.
this is one of those consequences of weakening unions is that part of the benefit to capital of a union is they can create these guardrails. if the great body workers all simultaneously stop seeing the union as having meaningful power enough to protect them from egregious abuses, it opens up wildcatting, sabotage, occupation etc.
ehat is distinct about public service in the austerity era is that few committed public servants wants to actually wreck shit or otherwise degrade public services, especially since that's what the political project causing the shut doen seems to want.
figuring out a tactic for sticking it to the politicians while continuing to provide value to the public is complicated and tends to be specific to the agency, and there usually tend to be laws against it.