People are often short termist
Which can be problematic with many geological climate related processes which are slow.
If we stopped any man made CO2 production today, we would not see an instant swift drop in CO2 levels, it would e slow. There would still be lots of ice sheet melting to contribute to rising sea levels (for simplicity will also ignore thermal expansion of ocean water as there's quite a lot of "thermal inertia" so we are not yet really seeing that effect of a slightly warmer climate yet )_
But because things are very slow, then politicians don't really care as the limits of their long term views are the next 1 or 2 elections, so climate change won't really get their proper attention until it reaches the situation where whatever changes are made will be too little and too late to prevent some real nasty effects..
The worries are not about total human extinction (though it wont be great for lots of animal & plant species, we are exterminating lots of those on a daily basis regardless of any climate effects).
.. Though could easily see a very small human population remaining and back to a low tech society.
If the expected happens e.g. major sea level rises, changes in major circulatory systems (be that ocean currents, the jet streams etc) then problems will start.
Just looking simply at sea level rises, a lot of important cities are based on the coast (for lots of reasons, e.g. historically fresh water from rivers running to sea, flood plains make for good farming, good fishing on rivers and sea, much trade was by boat)
Rising sea levels could see the "loss" of lots of major cities.
.. Let alone other climate changes such as previously pleasantly habitable & farmable areas, becoming too hot / cold / dry to be viable), though conversely some other areas may become viable.
Though there will (for a long, long time, unless CO2 and CH4 levels get super excessive) always be a few human friendly areas.
So we could easily see conditions for mass migrations etc... and if there's one thing humans are good at, its killing each other over such activities, climate change could give all sorts of "reasons" for war. If any of those escalate to nukes then climate change becomes less of a worry as wiping out big swathes of the population & destroying advanced tech will do a great deal to reduce CO2 emissions.
Even without major wars (.. as if..), our tech heavy societies could struggle with some of the adaptations needed (e.g. if in future we have moved heavily to nuclear power plants to reduce CO2 emissions,
as currently they need a lot of water they tend to be near the sea, which is fine, until sea level rises finally start to seriously kick in )
So, serious efforts to combat climate change would help our descendants quality of life (we are a selfish species, so here is a selfish reason)
Making climate change mitigating actions can be (relatively) inexpensive and low impact, given the nasty and massively expensive possible consequences, so precautionary principle would say it's worth going all in on climate change mitigation effects just in case. That's why we invest in the future even if we do not directly benefit, e.g. taxes fund schools, people without kids (generally, unless they are ******) don't begrudge those school funds as despite their personal no kid choice they know that future generations need educating for everyone's benefit.
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And if the fossil fuel company shills do inexplicably turn out correct, & climate change is a hoax, then not much has been really lost making efforts to keep CO2 and CH4 low