What is a pet in the relevant sense here?
Let's assume you have everything in the Culture except the ASIs. So you have full blown Drexlerian nanotechnology, access to all the technology more or less possible out to the limits of physical law. Energy is functionally infinite, and if you want basically anything to be produced you simply order it from the AI equivalent of ChatGPT today and it is directly manufactured from base chemical feedstocks and delivered to you by a simple automaton. Food, clothes, furniture, any sort of electronics. Homes are extremely plentiful and similarly require ~ no effort to obtain, since they are just snapped together from atomically perfect panels. When you're done with something, it gets recycled back to feedstocks to be made into something else.
Okay. Now what? You're free to love who you like, befriend who you like. But nothing you do matters really for anyone else beyond sentimental value, since nothing you do effects whether they live or die (you're all functionally immortal after all, plausibly you can backup your consciousness and get resleeved if something truly catastrophic was to happen to you). There's no scarcity, no need to horde things, you play games and amuse yourself in whatever little hobbies delight you, and you socialize with those who you care to.
What exactly is the grand thing that you think people are going to be doing in that scenario that they can't do in the Culture?
It seems apparent that for the overwhelmingly enormous majority of people the Culture is identical to the case where there are no Minds, because to whatever extent some tiny portion of humans needs to make some sort of major decisions in a post-scarcity world with no Minds, 99.9999+% of people will not be among them. We see mostly people interacting with Special Circumstances (a tiny fraction of the overall population) and most of those who live on GSVs etc managed by Minds in SC themselves are also just going about their days and not involved with whatever machinations the Minds have going on.
Like, what is the problem that ASI really introduces here for regular people? And in a world where you are indeed free to leave the Culture (as we've seen) if you really want to, and there's plenty of variety in how smaller groups in the Culture live their lives, what exactly is missing? There is no oppression, no brutality, no war, disease, famine, poverty, or want and you're free to do as you like except kill people without their consent and/or otherwise hurt others. What more freedom do you really want?