The crux of the matter is hidden in how the meaning of "caring" is defined. That's where opinions and perspectives diverge completely.
Here's a big question: How much change on a personal level are you willing to make to help curb climate change?
The trade offs and the compromises to be made are going to differ between groups and people. It all rides on social economic background, geographical location (where you live), mobility, opportunities, the life you've build for yourself.
Concrete example. Biden shutting down Keystone XL and the ensuing social upheaval:
https://www.vox.com/22306919/biden-keystone-xl-trudeau-oil-p...
In reality, several thousands of workers lost out on a seasonal / temporary work:
https://eu.statesman.com/story/news/politics/politifact/2021...
Disregarding the details of this particular case, it's an example that exactly points at the glaring challenge ahead: how this affects people directly.
Here you have a few thousand people directly having to figure out how to cover their living expenses, find an alternate livelihood to cover the loss of their job in the short run, adjusting their plans. This not a minor inconvenience, but quite an adjustment for many months to come. One can empathize on a personal level: losing out on a job - even if it's temp work - just sucks all round.
And that's just one of many uncountable individual trade offs people are asked or forced to make by their leadership. All risking to create resentment and lose support down the line.
In that regard, serious government officials and politicians do care, it's just that they can't make life-altering policy changes without consequences. The argument "If we don't change, climate change will end up costing us much, much more" is true to an extent, but hardly an argument you can readily use lest you want to lose support. People don't just look to you to enact policies regarding climate, they also look to you for leadership that ensures their lives remain relatively stable as well.
That's the catch-22 which humanity faces.