You have a link? The cost depends of course on how radically you want to bring down emissions - some targets may be unrealistically expensive, sure.
But 'some people will have a shitty life' seems like a pretty optimistic view of the havoc that climate change can wreak, considering that people who 'have a shitty life' generally aren't content to sit back and suffer in silence, certainly not when this is a direct consequence of the actions of other countries. If certain parts of the US become significantly less attractive to live in due to water shortages, declining agricultural returns, increased flooding, etc., it's still relatively feasible, though still disruptive, for the people affected to move to other regions within the US in large numbers. But when all or most of a country suffers such effects, and we need to look at relocation on a huge scale to other countries... that's a lot trickier.
As I was trying to say, I agree with you to some extent about the idee fixes of some of the left, their apparent refusal to consider that climate really does change also without human interference and their notion that somehow the climate as it was until recently is the natural one and everything else is a catastrophe. But nevertheless, it's clear that the climate IS changing and that this will have very significant consequences with very significant cost impacts. And it's also clear that there is a clear correlation between CO2 levels and global temperatures, hence it certainly makes sense to take major steps to rein back CO2 emissions.
Whether truly painful, radical changes are justified, the kind that cuts your GDP growth back by several percentage points per year, let's say, that may be up for debate, yes. But the steps advocated by the political mainstream left are nowhere near that level (obviously there may be radical groups who do advocate it).
The Little Ice Age certainly caused massive disruption too, but I'm not sure why that's a reason to not try to limit the disruption now, now that we can actually measure it and have a clue on how to limit it.