Like, people in the 20th century were living in constant fear of mutually assured destruction for dozens of years. Is that better or worse than the guaranteed collapse of the ecosystem? I couldn’t tell.
But these days, no one really cares about a nuclear war, it seems, even if one of the nuclear power is currently fighting a war; and it’s appearing to lose it.
It is successfully using its nuclear arsenal to deter aid for the country it's invading. A lot of materiel is sent, but its uses are limited. A massive attack on the capital would be entirely justified, and is well within their capability, but is forbidden for fear of escalating to nuclear weapons.
Because of that, they are continuing to prosecute the war, and making minor gains. They cannot continue that forever, but in another month, their target is likely to be hamstrung even further. They may yet win, or negotiate a peace that they can pass off as a win (which would include Ukraine never joining NATO).
People may not have that immediate fear of nuclear war that characterized the 50s through the 80s. But the threat is in play, and it's being kept out of daily discourse only because the US refuses to challenge it.
Maybe in two or three decades we’ll discover that Russia was nearer to using nukes than we assume.
The fallacy in your argument is that climate disaster is already happening. We have already rendered hundreds of species extinct. It is not a future risk.
Likewise in reverse, there are ways a nuclear war could be relatively good for almost every species except humans, our livestock, and our pets.
Given the trends on renewables, I think we're going to solve the climate… but also a whole bunch of seemingly unrelated trends start implying silly things starting in the early 2030s, so expect something else to become a problem soon enough.