"Our athletes are putting in the same effort, same training, and they deserve to be paid equally."
I'm more of a market forces person.
It's kind of obvious that a nurse "deserves to be paid equally" with a celebrity if we measure value on the basis of what people deserve. But celebrities get paid more. Generally-speaking, we don't do pay according to what people deserve or on some imagined idea of fairness. It's done at the place that markets say they should be.
If women's sports generate the same income that men's sports do, then I see no obvious reason for a different rate of pay. I'm not sure if that is true for women's tennis. Personally, I prefer to watch it.
But unless the income a sport played by women generates equal revenues, then I see no reason that women should get paid the same as men. Same in any industry.
Generally-speaking, men aren't as interested in reading clothing magazines. So male models get paid less than female models. It's not a supply side thing. It's about the scale of the demand.
But given your principle, what about the point I made previously. Why aren't women who play American football paid the same as the men? If you assume it's about effort and training, and not the income the sport generates, why shouldn't women footballers also demand equal pay with men?
For me, the principle fails because it doesn't account for demand.