As usual, BC brings up good points. But I am not sure about the phrase “without thinking”. I expect they thought about it more thoroughly than we have.
Consider this: Stanford allows its name to appear on ads for Nike. But then the next player does ads for a controversial organization (say, a cigarette seller, a political party, a religious affiliation, etc.). If Stanford prohibits “Stanford” from appearing on that after allowing it to appear on other ads by the athlete, how does that sound?
Someone could schedule athlete ads in such a way to make it appear Stanford takes a position on some product or ideology when Stanford doesn’t.
It is cleaner to consistently remove the Stanford brand from the athletes’ use.
Consider this: Stanford allows its name to appear on ads for Nike. But then the next player does ads for a controversial organization (say, a cigarette seller, a political party, a religious affiliation, etc.). If Stanford prohibits “Stanford” from appearing on that after allowing it to appear on other ads by the athlete, how does that sound?
Someone could schedule athlete ads in such a way to make it appear Stanford takes a position on some product or ideology when Stanford doesn’t.
It is cleaner to consistently remove the Stanford brand from the athletes’ use.