Neither of those statements are wrong (except that the sixth mass extinction event far predates the 20th century). But I think you're missing the point. The paper you linked is very much a consensus of climate scientists on the worst case scenario by 2100 if mitigation efforts fail. That scenario is very, very bad, and we need to address it with systemic change to avoid that happening.
But a huge number of climate activists are pushing the idea that the minimum possible effect is far, far worse than that. The effects described by that paper drive out humans and a lot of other mammals, but there's plenty of forms of life that can survive there. It won't produce a "band of death" where nothing at all survives. Far less the "total human extinction by 2100" that is popularly claimed, let alone the "Earth will be a lifeless ball of rock and water by 2100" that more extreme activists claim it is. It is possible that some of the people doing so were mislead by pop-sci (there has never been an actual consensus that things were that bad, but a lot of preprints got wide circulation before being withdrawn for serious underlying flaws), or thought that they needed to shock people into action, but that's probably not why so many of them are doing it now. Many activist groups are pushing a "NO HALF-MEASURES! ANYTHING THAT DOESN'T SOLVE EVERYTHING IMMEDIATELY IS BAD!" approach (Michael Moore's "documentary" Planet of The Humans attacks renewable energy and electric cars on this basis), and there is a growing "IT IS TOO LATE TO STOP THE CLIMATE APOCALYPSE!" movement. The effect of this is that it becomes incredibly easy to present the entire climate-change awareness movement as kooks that can be safely ignored.
This is "coincidentally" in parallel with the growing push from the fossil fuel industry and others to shift all blame for climate change on individual action (the "personal climate footprint") as a deflection from the serious institutional and systemic changes that are needed. A lot of the climate activist groups are dangerous as hell not because there is no climate problem, but because they're actively damaging the efforts to address it.