To be fair, wet bulb events are pretty much stage 1 of "this place is no longer habitable". Unless you mean "this place" as a opposed to "that place", then yes most people won't care until it happens to them.
I do suspect there might be a scare coming that will shock us out of that though. We will likely get to watch tens of millions of people in India and Pakistan die over the course of a few hot days at some point in the not too distant future.
I don't think it's binary, exactly. We have entered a grey area where large swathes or land near the equator is technically survivable but really sucky to live in, where one month of the year is spent sheltering from the heat in an air conditioned mall while you're barely working and your kids' school is cancelled, and the other 11 months of the year suck more than they used to but you're still surviving and working.
As we go from 2C to 4C warming that's when the mass deaths near the equator should start happening from the wet bulb temperatures due to sheer lack of AC in poor areas and households. But even then we might figure out logistics and information systems to get rural and poor people to the nearest mall when a heatwave is expected, which will be often. Assuming a blackout doesn't happen. They might not die if these systems work properly, but it'll be terrible for their economy and human development indices.
At 6C warming, it's just an even worse version of the above.
Whatever the case it's going to massively suck and the people who will pay the biggest price will be the people who have pumped the least amount of carbon into the atmosphere. Life is just structurally unfair.
