I own tons of white tennis shorts and almost all of them are see-thorugh. I really don't see what the issue is. It's rare that you can also see through your underwear, even if your underwear is white. And if you wear colored underwear and people can see it through your white shorts, so what? People already know you wear underwear under your shorts whether they can see it or not. And if they can also see through your underwear, well, what are they really going to see that they haven't seen before?
The pros don't seem to mind as most pros wear white tennis shorts all the time even though there are huge crowds watching live and millions watching on TV. How many people are watching you play tennis?