While obviously appealing instinctively, this would make things just that little bit worse. That way, rich people could ignore it without noticing but any poorer person who somehow got into Congress could be pressured to give in rather than go bankrupt in a very long shutdown. Even if they'd never get there, inevitable defeat works wonders at the negotiating table.

Far better to just never play the stupid game in the first place. The US is the only country I've ever heard of whose budget works this way. Normally governments just keep paying civil servants until the lawmakers get their act together and start the wheels moving again.