I've been a regular gym goer for quite a long time now and had several experiences approaching women in this environment and observed a ton of approaches as well.
From my personal experiences, none of them amounted to anything and they all resulted in a lot of unnecessary drama being created. I mostly blamed the women for this due to attention seeking behavior on their part. I think the biggest issue with gym approaches is that many of the women there are incredibly into themselves, it's the norm to see them constantly taking selfies in the mirror, etc. Women like this need attention and the gym is a perfect environment where they can get it. They may flirt and be open to talking with you while there but it's easy for them to leave it at just the gym. I feel that the "slow game" approach with something like this is an issue because you may start developing feelings of attachment to some woman you haven't even be dating. If she is a regular, I feel a couple casual interactions to get a feel for her before making a move is the best way to go about doing this. If she gives some excuse, flakes or is just difficult to spend time with outside of the gym I'd quickly cut her out and move on.
One of my initial approaches I just talked to for a few minutes and then went back to my business. She then approached me the next time she saw me and we were supposed to go hiking on this hill that has an observatory on the top, it's very public and tons of people around. Anyway, she ended up flaking on the date and ghosted me but then used me as a source for attention.
Another one of my approaches I just interacted with for a few minutes and asked for her number, to which she says that she had a bf. I then said "it was nice to meet you" and then left. However, this woman went OUT of her way to interact with me, flirt, etc. and again just used me for attention.
If you are able to filter women like this out very quickly you'll be totally fine. Also, if they aren't respecting boundaries, such as the case with the latter example I provided above, you need to be very firm with women like this.