Maybe her train of thought was for her to be your gym friend? Make you her orbiter? or have the privileged of both of you flirting with one another without any commitment nor s*x? Maybe a more nonchalant approach would have worked. Maybe she would have dropped her boyfriend for you if you played your cards right. Who knows, but in reality, the hit rate of dating medium to low interest girls is really low.
I knew someone that was able to date a gym girl around 2015. It started with smaller comvos, to them playing ping pong with us in the rec room, to him being able to take her out. It did take awhile to get to this stage, and I guess it helped he had a nice BMW, but since the girl was friendly, not cold, and not distant with everyone, anyone with enough game could have bang her.
This one I'm positive was an attention seeker. I mentioned this earlier in the thread, but a couple months after I stopped talking to her this other guy from the gym came over to me and asked me about her, as he had seen us talking before. Anyway, he then said that she did something similar with him. I'm sure that this woman probably behaved in a similar manner with any man that approached her at the gym.
I believe that the gym environment can be a problematic one for both parties if things line up in a strange way, like with my story. It's an environment where most of the people there are familiar strangers, meaning you recognize their faces but you don't really know them. You are highly likely to run into these people repetitively, so there is the possibility of awkwardness for sure.
In my honest opinion, there is nothing wrong with approaching in the gym but there needs to be some clear boundaries. If you approach a woman and she says "I have a boyfriend" and say "Nice to meet you" and go about your day that should be the end of the interaction. There is no further need for interaction at that point and there is nothing awkward about it. However, it becomes awkward when the woman starts behaving in a manner where she is not only encouraging further interaction but actively creating it.
Something else is that many women at the gym are attention seekers. Anyone that goes to the gym knows exactly what I'm talking about,
@SW15 is quite aware of what I was discussing earlier. These women are addicted to attention and the gym environment is the perfect way to gain orbiters. They can reject you but still have you around for attention where they feel comfortable in a non isolated environment.
To sum up what I'm saying, I don't think the gym environment is a great way to meet women and there is a degree of potential for it to backfire on you.