It’s not quite true. It’s very unlikely the planet will become completely inhabitable to humans anytime soon. There’s going to be a tipping point of enough extinction to completely stop any more damage and return to a balanced ecosystem. Once that happens, it’s very likely the people with the most power will be the ones in the remaining habitable zones.
I think your assumption operates on the premise that some form of recognizable modern civilization will remain. I don’t think it will. Once mass deaths start for humans it will also cease the global flow of petroleum products, materials, and foods, either due to wars or just the structure being so damaged that it cannot function. Once you can’t get fuel, can’t mine things, can’t refine them, can’t transport them, can’t fix the machines that make fertilizers or tractor parts, can’t keep the computers running, the internet collapses…that’s it. Hell, that’s the good scenario, not the one where the ocean overheats /-acidifies , killing everything in it including much of the planets oxygen generating algae. Civilization is done. Almost all surface minerals are gone because we’ve already mined them, well, except coal…but that’s what helped get us here in the first place. We are literally back to a Stone Age or scavenging materials from the bones of civilization.
Mad Max or some similar post-apocalyptic desolation is a far more likely scenario than any situation where a holdout of modern civilization exists.