I haven't seen any projected warming scenario where it'd be 31C everywhere. There's a gradual change in habitability as the habitable zone becomes smaller and smaller.
Perhaps our descendants will be farming Antarctica.
Sure, but will humans just sit idle by and wait for extinction, or will they do whatever it is necessary to keep the ecosystem from boiling away? Of course it depends on how severe the problem is, but in general we have the means to affect climate. (Solar shade comes to mind and large scale GHG capture.)
You're right to a point. The hotter it gets on average, the faster the less fit die due to various heat related reasons. Once it hits 31c@100% relative humidity, land mammals die.
So it's on a rolling scale up to 31C. At or past, yeah its a binary cliff.
Even the worst case models predict that Earth will be habitable for humans for a long time. IIRC the worst that can happen in the next 100 years is that the sea will raise a few foots and flood some costal cities, change the sites where you can cultivate some food and cause hunger in some places, more floods and drought in many places, and kill many animals and plants. The worst case prediction is a huge huge huge disaster but it is still faraway from the point it is uninhabitable for us (and uninhabitable for roaches (and uninhabitable for bacteria)).