With photography, the solution is exactly what you specified: you use the anti-flicker shoot. Most good cameras have this. Some entry-level cameras may lack it.
There are many kinds of LED lights: some flicker, some don't. A non-flickering light presumable uses some sort of electrolytic capacitor as an energy storage component. Such components fail early and often. A flickering LED light may be able to omit that capacitor, which not only lowers the cost of those LED lights, but also makes them more durable.
According to quick googling, an average gym is about 4000 square meters. In my house, I have about 70 square meters of area where lights were not initially installed (and I have to buy the lighting). That lighting cost 630 euros, and to me as a photographer is not "good enough" so using flash is mandatory. If you increase light levels 3-fold, and increase the area 57-fold, then I assume the gym would need over 100 000 euros for the light system. And with that money alone, you can't be certain that you get flicker-free lights.
My advice would be to just focus on high brightness, good color rendering and optimal and consistent color temperature and use those as specs to buy the lighting system. Those features, especially high brightness, are what professional photographers are looking for. Professional photographers already have a camera with flicker-free shooting so that's not on the list. A camera with flicker-free shooting costs maybe 500 euros more than a camera that doesn't. If we assume the cheapest adequate flicker-free system would cost 150 000 euros and the cheapest adequate non-flicker-free system would cost 50 000 euros, that's 100 000 euros difference and would make sense only if more than 200 photographers shoot in the gym during the lifetime of those lights which may be as little as 5-10 years.
For someone considering buying a camera, it's easy to see if it has flicker-free shooting as an option. For someone considering buying LED lights, it's not easy to find out if they are flicker-free.
It's possible to test, though, if you are making a large purchase. If you want to purchase 500 euros of lights, it may be tricky to arrange a test shoot session before buying. But if you want to purchase 100 000 euros of lights, it may be possible to arrange a test shoot session for many candidate lights and buy the one that doesn't flicker. But my advice still would be to let the camera buyer take care of that by buying a camera that has flicker-free shooting.