I was thinking that a 'search for Kirk' kind of movie wouldn't be too bad if the alternative is no Kirk or recasting. Pine would get the paycheck for a smaller, but still very important role.
Spock can be the acting captain, Uhura and Sulu the competent officers who join efforts with him. Bones would be conflicted about his friend vanishing (Kirk's importance to Mccoy, and their friendship, is as underrated by fandom as the spock/uhura bond or the kirk/uhura dynamic. The guy would be destroyed if he lost his friend, more than Spock would honestly. Because Bones has no one else )
You could have a different movie where the captain has gone mia and the crew has to find him. Explore what happens in such circumstances. Add some mystery around the planet or section of space where he vanished too. Make things unpredictable. It could be an opportunity to do more with the other characters too.
Those who say that the general audience wouldn't care about a movie focused more on Spock: I beg to differ. Because, like JJ&Co admitted themselves, the first movie was about Spock more than Kirk and it was successful. There were many people saying that the movies should've made Spock the captain, or focus more on the aftermath of the Vulcan diaspora than Kirk's daddy issues. Many were expecting a bigger focus on Spock. You can't negate that his character is strong and popular. It's Spock.
If anything, it's Beyond (the weakest of the 3 in terms of success) that sidelined him, in a way, but in the first movies he was co-protagonist with Kirk. Pine never really was treated, in a sense, like Shatner was back in the day.
That was part of the reason why the original trio wasn't the focus too: because while the original trio was first foremost a dynamic about Kirk and the devil and and angel at his sides, in the reboot Spock is elevated to a different role than being just the nerdy, logical friend of hero. There's less of that one dimensional, characters having symbolic roles, approach in the narrative too. In fact, elevating Uhura to the original trio level was itself the result of Kirk and Spock being more equals here, so Spock is allowed to access to narrative elements only Kirk would be allowed to access, and Uhura herself is allowed to access to the part of narrative (the interpersonal relationships with the main guys) Mccoy is allowed to access to with his role as Kirk's friend (it should be noted that both Uhura and Mccoy represent dynamics that already existed before the enterprise and are thus a constant for k/s. They are the continuation of their backstory from the academy)
To be honest, Lin tried to restore the original trio in beyond but in the end, he still didn't and they don't really have so much focus when I think about it. The problem, if you consider it one, is that it can't be the same dynamic because this Spock is more contemporary too ( Urban himself finally seems to have realized that in an interview, and thus understood why the old characters' banter would be forced here and Mccoy cannot have the same exact role he had in tos because Kirk and Spock ARE different. Hence, it isn't like the previous team wanted to sideline Mccoy, they just need to make him part of a different narrative and they couldn't keep things the same as tos because it wouldn't align with different purposes their own narrative has )
Tl dr: they need Kirk, but if he can't be front and center and we don't get a movie about his daddy issues, it isn't the end of the franchise because they had already tried to set up this trek in such a way that if the next movie focused more on Spock and the ensemble, rather than being the Kirk show, it wouldn't be a tragedy.
And honestly, this whole bringing George Kirk back wasn't really a matter of them doing what the audience was lobbying for as much as it is just an attempt to use the popularity the actor has now thank to Thor. Of all the things the next movie could be about, I frankly doubt Kirk's daddy issues and time travel are at the top in the list of things the general audience wants to see, or expects to see. I don't think people will have a nervous breakdown if they don't use that story.