I think that all of the answers are great. In my experience perhaps the most significant reason is that your client is concerned that others in the gym are watching him/her. Your big challenge is to help your client understand that the others in the gym are there for their workout, not to watch others, except maybe in Beverly Hills! Focus on the workout, not extraneous observation.
People prefer to work out with those who are the same boat. If they are out of shape or unskilled at certain exercises, they will feel out of place among a minority of regular gym-goers. They would be much more comforted to work out in a private setting or in a gym with mainly out of shape people.
The list can be as varied as the people. Here’s a few off the top of my head:
-Past bad experiences
-Not sure what to do when there
-Think everybody else is “in shape” and knows what they’re doing
-Lazy (yes, I said/typed it, there are lazy people in the world)
-Think they don’t need to
-Would rather play outside and/or with a sports team