I really like “the great extinction”

More punchy. More descriptive of actual events.

“Climate drift” sounds like the name of an AI bot, made specifically to create a narrative that underplays the seriousness and gravity of the extremely dangerous and deadly feedback loops that are appearing now in the global climate system.

Drift carry the idea that we cannot change it back.

I don't think it underplay anything. I understand that we must act. But with this name your arguments won't be heard.

The climate denialists of today are those believing that if it gets too much, we will have to do some efforts for a decade and boom, back to 1970 climate. I want to engage that belief, hence 'climate drift'

The implication of “great extinction” is that we’re seeing a mass extinction event which we aren’t. At least not on the level of the major ones usually discussed.