It's certainly possible that a renamed SHN would be more likely to bring in popular shows that usually run in nearby cities (Wicked, which is playing in San Jose this week, for example). But I'm just guessing. They are more likely to program as they have been doing, when first-tier Broadway tours are on their West Coast swing along with longtime favorites to fill out the season. The only benefit for SHN of Harry Potter occupying the Curran for a long while is that it will basically have a monopoly on Broadway tours anyway. (I would, for example, think the Hadestown tour would be a better fit for the Curran, but the Curran will presumably still be occupied.) The Hays-less SHN can even get a tryout or two that otherwise would play at the Curran or Berkeley Rep if they're a little daring.
The Curran is Hays' baby, so I assume (in total ignorance; I don't know the arrangement and could be wrong) that Ambassador Theatre Group will be there to run Harry Potter and nothing more. The Curran has had trouble filling out a season and drawing a crowd, with the obvious exception of Dear Evan Hansen and partial one of Fun Home. The big question: What happens when Harry Potter leaves? Will the Curran be a strong competitor for the most popular touring shows, as it was for DEH (which, as a smaller show, made more sense in the Curran anyway)? A Curran season that has some popular Broadway touring productions along with out-of-town tryouts and experimental fare would make more financial sense while appealing to me. But Hays has yet to prove she can program and market effectively despite having the nicer theater in the better neighborhood. Basically, she has provided a service to a cheapskate like me to see shows from Bright Star to Head Over Heels to The Jungle for rush ticket prices and excellent seats (unlike SHN, which has taken to putting rush ticket buyers in the mezzanine).
I can imagine the new SHN dominating the scene and getting most of the popular hits, leaving Hays, post-Potter, in a weaker position than before (when she was collecting SHN profits). But I can also imagine a better-managed Curran putting together a strong season that mixes the popular and the daring more effectively than before, and stealing even more touring shows that fit better in a Broadway-sized house.
But it's going to be a few years before we find out, I guess.