Most of those "rights of publicity" don't apply in the US though. If they did, half the actors in movies couldn't get work, because quite a few of them get work as the off-brand version of X in the first place.
Off the top of my head I can't think of a single example of any actors that come across as an "off-brand" version of someone else. What are some of the examples you have in mind?
I don't think this has anything to do with this case, but there certainly are some "types" in Hollywood with doppelganger actors.
Isla Fischer / Amy Adams
Margot Robbie / Samara Weaving / Jamie Pressley
Justin Long / Ezra Miller
Not to mention all of the token/stereotype characters where it hardly matters who the actor is at all. Need to fill the funny fat lady trope? If Rebel Wilson wasn't available, maybe they can get Melissa McCarthy.
The voice from Her isn't even the first voice I'd think of for a female computer voice. That trope has been around for decades. I'm sure OpenAI just wanted SJ specifically because she's currently one of the most popular celebrities in the world.
Another one you didn't mention: until relatively recently, I thought actors Elias Koteas (Exotica) and Christopher Meloni (CSI: Special Victims Unit) were the same person!
This is where your post breaks down. Many people say they don't think the voice sounds like SJ. Others do. But it appears you've made up your mind that they deliberately emulated her voice?
It protect celebrities who rely on endorsements and “who they are” for income.
It very clearly prohibits copycats with near-likeness as a workaround to getting permission from a celebrity.
OpenAI asked SJ to use her voice. That right there helps her case immensely.
She said no. They went ahead anyway, presumably with someone or someone’s with a similar voice.
They publicized the product by referencing SJ.
These facts are damning.
They might be just a part of the story. Maybe 100 actresses, all sounding roughly the same, were given the offer over a two year period.
Maybe they all were given the same praise. Maybe one other, who signed an agreement, was praised on social media much more.
But this isn’t a slippery slope or a grey area. SJ was asked and said no.
That prohibits using a similar sounding copycat and publicizing as SJ.