Continuity of government is kinda the #1 priority of government as everything else they do is subordinate to that.
Well, to be fair, anything that is considered "essential" for the government to do is not, by definition, shut down.
Obviously, there will be differing definitions of "essential", but the continuity is there.
That said, amending the appropriate legislation to prevent this nonsense from being necessary when branches of government are at a stand-off should be the desired goal.
I like the 'triggers an election' thing. Puts the right pressure in the right place - if people in place can't get along and come to a reasonable compromise to keep the country working, then the people need a way to make some changes.
https://en.wikipedia.org/wiki/Continuing_resolution#List_of_...
Deadlocks like this are definitely a feature of our system that Westminster-style systems resolve differently, but shutdowns are an artifact of an old law more than the constitutional organization.
(ditto the "debt ceiling", which only exists because of a stupid statute)