I think this just varies from person to person. A friend of mine brought her young baby to the gym with no qualms whatsoever. My daughter is 17 months old and I'm just getting to the point where I'm thinking it might be OK. I don't think there's a right or wrong here. We have instincts not to leave our helpless infants with strangers for a reason, but I don't think evolution was thinking about the child minders at the Y, you know? As for getting sick, he's got a 3 year old brother. 3 year olds are not exactly fastidious hand washers or careful about what they touch, so he's probably getting all kinds of germ exposure at home.
If you feel you would benefit from some me time, why don't you just try it? If it's unbearable, put your street clothes back on and go get him.
I think it would depend on who the childcare providers at the gym are. Do they look experienced? I might ask them some "what if" questions. I was pretty trusting about this when my EBF DD was 6 mo old, and the childcare workers gave her peanut-butter crackers.
and then when we get to the ocean
we're gonna take a boat to the end of the world
I think it depends on the gym. I tried two different gym day cares. They were fine for my older kids, but I didn't like them at the baby stage. In both day cares I had problems with the workers not changing my kids if they had wet diapers. Also in the day care at the Y, people brought in their sick kids all of the time, with everything from fevers to pink eye to hand, foot, and mouth disease. I know because I have a friend that works in one of the Y's (I would trust her with my kids anytime, she has done daycare for years and is very responsible, but a lot of the younger teen workers are not). I would think that there would be some kind of "three strikes, you can't use gym daycare" anymore for people that constantly bring in sick kids, but at the Y my friend works for they really aren't allowed to say much to the parents.