Actually quite a few average people somehow play pro sports. You know, Lebron James, Ronaldinho, Paul Pierce, Barry Bonds, Payton Manning etc

There would not be an even balance. Have you watched high school sports before? 

 Not extensively, but a bit. The level of physical demand is much much lower at that point. It is still easily conceivable for someone to push the extra mile to get better than someone else and make a spot on the team. It is competitive, but not a super cut throat situation. The differences between the sexes is rather small on the whole so it wouldn't be as big a factor when dealing with a limited enough pool of people. It would only really become an issue if you can pick and choose an entire team consisting of the top of the top athletes for the sport.

I have watched high school sports rather extensively, and I played football and also wrestled. The differences between the genders may not be as pronounced as in the pro leagues, but it is still pronounced. Granted, you would have the occasional female standout similar to Britney Griner or Candace Parker make a boy's team, but the vast majority of female athletes would not. 

This is all a moot point.

There are not laws or restrictions against women playing in professional sports. The truth is just that there are not any women out there who could possibly hope to compete with men at that high of a level. I'm talking about "real" sports (this will probably start arguments lol), not golf, racecar driving, table tennis etc. The only exception that comes to mind is Billie Jean King in tennis. But today the gap has widened considerably between men and women with the onset of steroids and other supplements.

Is it not fair that women don't then have their own leagues? I think it's perfectly fair. Sports are a business just like any other business, and the product they sell has to be a good one. Women are, from that perspective, not as good of a product, so a lot of times there is no market for professional women leagues.

Your definition of average is quite laughable.

The average man is 5'9.

There is nothing average about any of those guys.  They were all born terribly gifted atheletes who worked hard to condition themselves even more, and to learn the skills to play.

Yet those leagues still exist.

The WNBA has been nothing but a giant sinkhole of money.  Women's college sports are pretty much never cost effective.  Instead taking money from the more expensive profitable men's programs... (and often complain that they don't get equal money and scholarships.)

The free market doesn't come in because people think that seregation is being fair for some reason.  Seperate but equal so to speak.  (Or seperate but not equal.)

Some sports, ok. Auto Racing, Motorcross, ok.

Sports in general are intentionally sexist since the primary audience is males. Everything from the advertising surrounding sports, the sports channels themselves, the cheerleaders, and just about everything else sports related is sexist.

So its not so much that female sports are sexist, its that ALL sports in general are sexist.

I think it's because sports are supposed to be fun, and generally when adults from both genders group together for anything other than drinking, it's not fun... even then that's not really fun either, more just a formality before sex.