I’m having trouble buying that the shutdown is a result of incompetence.
The GOP has shown us that they want to destroy the government’s ability to perform certain functions. Those functions mostly consist of pro-social programs. As part of their strategy, they try to reduce the population’s faith in the ability of the state in general. To do this, they do what they can to make the state look as useless and incompetent as possible.
A month or so ago the right wing in Britain was making waves with another push to reduce or kill the NHS. First they cut the funding. Without adequate funds to operate well, the NHS failed to operate well. The right is saying now the NHS is just fundamentally broken.
The right in the US appears to be doing a similar thing. They act like imbeciles and kneecap every effort to do anything useful. Then when the people doing useful things struggle to do so due to obstruction, the right blames the attempt to do useful things.
In this particular case, the GOP could keep the government not shutdown if they wanted. A shutdown makes people think the government is fundamentally incompetent, though. And people who think that will turn against things like universal healthcare or any other pro-social program.