The biggest thing that no one has mentioned so far with gym dates is the membership aspect. It's not that likely that you and some female you meet from either a random in-person approach or a swipe app will be a member of the same gym. That will cause logistical problems.
A possible scenario is that both you and the female are both multi-club members in big box gyms like LA Fitness or 24 Hour Fitness. That would enable a gym date to be possible with fewer logistical issues.
A regular gym date faces the issues
@Rainrain mentions if it can even be scheduled. Some gyms do offer fitness classes and that might be a better option for a date. I'll explain more below.
If somehow you can work past logistical membership issues, going to a gym class at a gym could work. It might be challenging to agree on a class format to attend together. If you can, it would be a decent first date but this is far less probable and far more challenging from a logistics standpoint than either setting a date for drinks in a bar, or even activity dates like walking around a park with a lake or playing mini golf/TopGolf. I'd rather schedule something easier logistically, which is why I tend to schedule bar dates as my first dates most of the time.
There are also standalone class-based fitness studios for some fitness formats. It's possible to go to a standalone studio for classes too. Since you mentioned cycling classes and I have some experience with cycling classes, this is how it would happen.
Cycling classes on stationary indoor bicycles are rather basic. Most generally fit people can do them at least semi-competently their first time. If you go to a standalone studio like SoulCycle or Cyclebar (two big national cycling chains in the USA), you have bicycles with specialized clip in shoes. Dealing with the clip in shoes for the first few times is annoying and takes time ahead of class. It's generally recommended to arrive early. A lot of women have issues like being on time for first dates, so I can't imagine that the typical female from an approach or an app is going to take this seriously.
The cycling classes themselves are about riding to the beat of the music and often doing choreography type moves in there. Most guys don't excel at either of those. There are some cycling classes that aren't choreo based (generally ones at places like LA Fitnes), but Soul and Cyclebar are. Other small standalone studios would likely be choreo based.
Getting seats in SoulCycle or CycleBar classes can be a challenge too. Most seats are taken by people with memberships or those who have bought short term class packages. As a guy, you would need to have some short term class package to book your own seat in a class and then get the female in on a drop in provided she's not a full member or class pack member at those studios. Additionally, even if you could get into the same class, it would be unlikely that you could get 2 bikes next to each other. That's a bit silly for a cycling class.
Some HIIT format class or yoga without designated seating is probably better than cycling for a date, but even those classes have logistical issues.
I'd recommend sticking to bar dates or a few basic activities like walking/local sightseeing or maybe an intriguing museum. There's a reason I do bar dates for the majority of first dates.
