Even the government shutdown is an example of the failure of the US constitution. In most other countries in the world, the inability to pass a budget triggers an election.
In many parliamentary systems, and some semi-presidential systems this would be likely or automatic. Basically no presidential systems do this, though some have optional safety valves that might be threatened to try to force resolution of a budget deadlock or invoked in response to one, like the muerte cruzada system in Ecuador (which allows the President to dissolve the National Assembly or the Assembly to impeach and remove the President, but either action triggers new elections for both the President and the Assembly.)
I find very strange that the presidential systems claim to be "republics", presumably because most people do not know what a "republic" had previously meant.
The most important principle of the Roman Republic was that it should not be allowed for any important civilian executive function to be occupied by a single human. All major functions should be held by 2 or more humans with equal power, e.g. 2 consuls for the supreme function (because it would be less likely for all of them to agree to do something abusive or illegal).
Only in exceptional circumstances, like war or natural catastrophes, it was fine to temporarily delegate power to a single dictator, to ensure fast decisions.
This principle of the Roman Republic was intended for avoiding the abuses of power, like those typical for kings. There is very little difference between most presidents of countries with presidential systems and absolutist monarchs, even if their function hopefully is not hereditary, but even this is not always true.
It is perhaps a consequence of some of the ways in which Presidential systems are suboptimal, but I don't think it is itself a way that they are bad. If you had a Presidential system and changed nothing else but making failure to pass a budget on a set timeline an election trigger, it would make things generally worse, not better, except maybe if the regular election interval was intolerably long to start with (in which case it would maybe be incidentally weakly, indirectly positive.
That's actually a wonderful idea. Our legistlature definitely needs more skin in the game, so I was privy to the idea that their salaries are also frozen during a shutdown (like 90% of federal workers). But having them ousted from their seats can be interesting.
Of course, the obvious counter argument is exploitation. Could a bitter party band together and try to force early re-elections if they feel they have the upper hand?
But, there was no abdication. There is an intentional cooperation.
Likewise, the Supreme Court putting an immunity clause right before Biden left feels like abdication. Again, if what Trump was doing was just, it'd be easy to interpret it in his favor, 6-3. But instead they gave blanket immunity. It can be intentional cooperation and still be abdicating.