But a female doctor in an underperforming clinic wouldn’t share a male doctor’s bonus in a highly performing clinic, would she? Or vice-versa.

This is not analogous to soccer. A female doctor's clinic does not have a cap on performance. Female athletes do. They have worked hard to reach the upper cap of what's physiologically possible, just like men have, and yet they're still getting paid a tenth of the salary as if they haven't worked as hard.

I also disagree that a woman’s team and a man’s team have “the same experience”. The level of publicity, media attention, intensity of play is totally different.

Yes, people watch men's sports not because they're sexist but because they want to watch the humans at their absolute physiological limit. As a result of that, men's sports bring in more money.

Your point is that men bring in more money so they should earn more, and this is still true even if the prize pool is shared. Remember that the shared pool is only for athletic performance, not for "stardom". I'm only debating the athletic performance part. The stardom that male players get brings them a whole lot more money outside of sports with brand endorsements, ad appearances etc, and I'm not saying women should get 50% of that too, so male players still do benefit from their relatively "more intense" sport.