> But it seems like certain jobs are funded for the duration anyway (infamously, members of Congress are one).
I think thats due to the 27th Amendment [1]
> No law, varying the compensation for the services of the Senators and Representatives, shall take effect, until an election of Representatives shall have intervened.
can't change (or stop) congressional pay until an election. guess it's a double-edged sword they can't give themselves in immediate pay raise, which I think was the point of ratification in 1992, but also can't cut their pay for failing to pass a budget.
[1] https://constitution.congress.gov/constitution/amendment-27/