I think I might have some additional insight:
It looks like if the fluid temporarily runs out, the machine will stop (normal)
But inserters will not insert into a machine with no fluid, unlike a normal assembler.
But an inserter will not update when the machine gains fluid again - it will still think the machine is empty until it is refreshed (source chest gains an item, inserter is rotated, etc.)
I'm guessing the machine has an auto-recipe based on the input fluid, and no fluid acts as no recipe. But the issue is that inserters won't insert into an assembler if there's no recipe, and the auto-recipe doesn't trigger an entity refresh. So it gets stuck.