Gotcha. Well, some of your diagnosis can be done quickly, so the belt could come off. But to really take some time with it and play around, just leave it all connected and let 'er rip.
Make sure the fan clears the shroud of course. If not, either remove the shroud, or, as you suggested, the fan.
Personally, I'd take the shroud and leave the fan to do it's job. But whatever works for you.
Good luck tracking it down. And be safe.
Paul