The biosphere will take it all in its stride

Well, not this particular biosphere. Whatever can evolve to whatever conditions end up stabilizing will recover...although recover isn't the best word since it implies returning to the previous state. The Holocene has already been declared over, we're just in an abrupt transition now. RIP to everything else that the humans dragged into the gutter with them in their attempt to be a higher form of life.

Life will come back, maybe in similar levels of a new epoch of diversity in a hotter world, however given the time scale I think humans may have not only ruined their own environment but condemned Earth life to be definitely now tied to the lifetime of the Sun, as no life is ever going to get to a level to escape its final form.