1997 Mitsubishi Eclipse Rough idling

I have been having problems with my car overheating and leaking antifreeze, so I took it into the mechanic. They ran a diagnostic test and it was the water pump so we had that replaced. They suggested that we do the timing belt at the same time as the water pump since a lot of that labor is the same however it was going to double our bill, and we could not afford it so we said no.
When I picked up my car, it was idling extremely rough. When I told the manager about this he said that the problem should straighten itself out within a day. A day went by and I'm still having palms with a really rough idle. When I called the mechanic back another manager answered and explained to me that the timing belt tension with off because I did not replace it when I replaced the water pump. He told me that the tension pulleys are now bad because they had to be taken off and put back on but were not replaced. I was told by the manager that this would cause the part to not function correctly.
I am wondering if this is a legitimate issue or a mechanic screwup that they are not wanting to fix. Can replacing my water pump but not my timing belt cause The tension pulleys to function incorrectly.