Sorry you're wrong. Zuckerberg did not want his name on the hopstial. SF General pushed him to put his name front and center (ostensibly so other rich people would be "encouraged" to donate as well).
The linked article says removing the name would require returning the money per the terms of the contract and "the hospital agreed in 2015 when it accepted the $75 million that it would keep the Zuckerberg name for at least 50 years."
Another article [0] says "[...] Zuckerberg SF General Hospital, a name the Zuckerberg-Chans bought themselves in 2015 for $75 million."
[0] https://sfist.com/2020/07/24/more-supes-call-for-stripping-z...
Anyway, here are two Bay Area journalists who discussed this a few years ago when this topic first came up:
I still find it curious that according to the contract, they would have to give the money back if they change the name. At least thats whats being reported.