They do. Timings vary from attraction to attraction but cast are supposed to take show quality rides during operating hours.
Pirates only did it right before opening in the morning... And the manager would get upset if something "minor" was marked not working because they would have to report it and delay opening, affecting their ratings.
(Like the mist screen projection)
Fun fact... Did you know that as long as TWO out of the three musicians are working, we're allowed to open the ride? (-:
EDIT: I can't provide much info for other attractions, but I had to do a checklist/ride-through for pirates.. It could give an idea of what is considered "important elements" for an attraction to have to close down if necessary.
List of essential animatronics/effects (at the time I was there)
1). ALL Jack Sparrow Animatronics are
2). Projection mist screen
(no longer there)
3). The interrogation pirate at the well
4). The auctioneer/red pirate
5). The captain in the fight scene
6). The pirate with the map
7). 2 out of 3 musicians
8). The lightning/picture effect
There's more but that's off the top of my head. If any of those aren't working, the ride has to close briefly for maintenance to fix it (if it's reported by a guest, and if the CM actually reports it to management)