Nike only exempt? I don't think so. I think it's a matter of having a so called "clean contract" (sponsor forbidding player not to display other brand logos on their outfit). Nike does have a tradition of forbidding all players including scrubs, from making extra money that way.. Other sponsors are more flexible (they do that only with some of their players.
What you say would make sense except that it really is about avoiding fines. In the case of a lot of other things to do with clothing sponsorship what you are saying is absolutely true but this aspect is a little different.
If you read the article above about Venus, you'll notice that Reebok, her sponsor, did argue that the WTA logo would violate their contract, but the wta still fined Venus and threatened exponentially growing fines if she continued to not wear the patch, and the only way to avoid increasingly larger fines was to wear the patch.
"WTA rules require players to wear the logo as part of its sponsorship contract. Only Nike athletes are exempted from the patch requirement because of a company policy that predates the WTA deal.
Reebok, which has a multimillion dollar sneaker and clothing contract with Williams, designed a seven outfit wardrobe for her at the Open and defended the logo-less look, arguing that wearing one would violate the company's agreement with her."
"Williams faced a $500 fine if she did not display the patch Thursday. The fines would have escalated to $25,000 had she continued to pass on the patch."
Therefore Venus was forced to wear the patch. Sharapova, on the other hand, was never fined.
So the point is that the wta is willing not to fine Nike players. Of course the wta logo is actually against the clean contract agreements of many players, but they have to do it to avoid the wta fines.
That was over ten years ago so some renegotiations may have happened with other companies since then, but this seems to be the historical basis of the Nike players not needing to wear the patch.