So how do you play the governing game when the administrative state is isolated, protected, and not accountable to the citizenry. Vote in a new president? Pointless, as far as the swamp goes, since a new president who, according to you, is powerless to drain even a drop of swamp.

Hard work.

There's no magic bullet. There's no way to "play the governing game" that only involves taking less than an hour out of one day every four years to vote in a Presidential election. That's the fallacy - the belief that all it takes is one election to make significant changes to the federal government.

You would need to go out there and convince lots and lots of people that they should prioritize government reform as an important issue in electing not just the President, but Congressbeings as well. A movement. You'd have to devote a lot of energy not just to getting people to care about this issue enough to vote it, but to care about it more than other issues. That's going to be hard to do, because most voters will care about other things far more than the Civil Service and government personnel reform, but that's what you would need to do.

BTW, most government agencies are very accountable - they're just not accountable solely to the President, which is what seems to tick a lot of people off. They serve two masters (to oversimplify): the President and Congress. Few agencies could withstand pissing off both the President and Congress. But they frequently find themselves between a rock and a hard place, with Congressional leaders (especially leaders in their respective Committees) that are pushing the agencies hard to do some things, and the President trying to implement his agenda through his appointees.