Plenty of things in climate-forward policy actually improve everyone's quality of life, even if the "way of life" changes.
Take EVs for example. As we transition to them, local air quality improves, not just for the EV drivers but also for non EV drivers.
Same for heat pumps replacing fossil burning furnaces - they are so efficient that it still results in a CO2 and emissions reduction if we burn natural gas at a power plant to power them.
And for most people and drives EVs are just a better experience (acceleration, vibration, smoothness), and heat from a heat pump feels better than a gas furnace since it doesn't excessively dry the air, and also has no risk of leaking combustion byproducts into your living space.
But sure, if "way of life" means the "principle" of burning fossil fuels directly for day to day propulsion and space heating, then yes climate policy is a threat to that, but being wedded to a technology for its own sake seems backwards.
Believing nothing should be done about climate change does not make one an extremist. Believing animals should not be eaten does not make one an extremist. Adhering to those beliefs by automatically discounting contravening evidence makes one an extremist.
The problem is that this is the religious position, not the other one. If you go investigate this properly you'll find there's tons of scientific dispute about even very basic things like what the temperature actually was 5 years ago in specific places, let alone more complex topics like what the climate would do in response to an immediate doubling of CO2 levels (so called ECS).
And the impact of global warming itself isn't at all obvious. All the effects so far are so small you can't directly detect them with your own senses, you have to use sensitive monitoring equipment and/or accept a lot of conjecture about chance and probability. In turn that requires you to have absolute confidence in the people doing the measurements and processing the data, but plenty of people don't have that confidence for various legitimate reasons (e.g. the Climategate emails). So it's pretty much by definition not obvious.
"Believing nothing should be done about climate change does not make one an extremist"
Doing nothing is the default position in any debate so arguing for it by definition makes one not an extremist. You can't be a "nothing extremist", that's a contradiction.
Extremism takes the form of positing extreme scenarios like the end of the world, and demanding extreme solutions like massive changes to everyone's ways of life. The people who say "let's not do that" are the ones pushing back on extremism.
This is at the root of a lot of the fighting over climate change: it creates enormous cognitive dissonance in people. Those who decide they want to fight it want to see themselves as rational devotees of science, but find themselves being the guy in the sandwich board yelling "The End Is Nigh" and demanding extreme policies. And just like sandwich board guy they struggle to explain the details of how the world is meant to end. Very few people who have tied their identity to climate extremism can actually handle a debate about the details which is why they instantly retreat into screaming that it's all "settled science" (which is a lie).
In particular, most people recognize the possibility (at least theoretically) that someone can be both an extremist and also correct. Whereas by your definition that's impossible.
Are nuclear power plants being shut down for climate policy reasons?
The above assumes that the proposed policy would actually work too. Some climate policies are clearly not going to have a significant positive effect, but we're doing them anyway (eg banning plastic bags in favor of paper bags@). Other policies could be designed just to keep yourself in power (deny the opposition power).
@ this does have a different reason too - to fight plastic pollution, but climate change is also cited as a reason. Paper bags are heavier and require more resources though, which makes it difficult for them to end up as better than plastic bags when it comes to climate change.