I can choose to believe that unicorns actually exist, fart rainbows, and lay eggs that hatch into a unicorn larval state that is identical in composition and taste to a gummy giant cola bottle. And I can choose to believe that no matter how much contradictory evidence you provide. Meh! You're lying! Fake news!
When people argue that we can't affect climate, it's a belief. An opinion. And certainly one for which mountains of contradictory evidence exists. The reason they believe this is because it's easier to ignore a problem if you convince yourself you couldn't possibly be responsible for it. No, no, we're too weak and feeble compared to the almighty power of Mother Nature to affect the climate ... or some such opinion based on nothing more than 'what I choose to believe'.
As for nuclear weapons ... they don't have nearly enough power to destroy something the size of the Earth. They'd just produce a few craters. Sure, they might kill off a lot of life either directly or indirectly through a nuclear winter, but bacteria would certainly survive (we've found bacteria happily living inside nuclear reactors!) as would many deep sea creatures, animals living in remote areas, or in caves.
Generally, though, people aren't stupid. We're just very good at convincing ourselves we are right and other people are wrong, and that our abilities are superior to others (in this case, assessing the probability that we are changing the climate). It's why something like 70% of drivers say they have above average driving ability!