It's a bit more than that. What you're missing is that without a healthy planet, there is no humanity Every eco-system on this planet is aligned to promote the current forms of life.
If they vanish, which climate change will promote, then new forms of life will emerge, but humanity will disappear.
The death of plankton in the sea, due to warming seas, is just one of them. There are multiple ways in which humanity will eventually die out - i n about 80 years.
So, no, it's not just a matter of people will move into places where it was previously uninhabitable because it was too cold, and the super hot areas will die out.
There is also the production of food. As honey bees die out, the means of pollinating plants go with it. As pollination dies out, we have no food.
As the surface heats up, the sub-strata heats up. As the sub-strata heats up, there is more volcanic activity. As there is more volcanic activity, there are more pyroclastic clouds. As there are more pyroclastic clouds, it blocks out the sun. As the sun is blocked out, there is no more photosynthesis. When there is no more photosynthesis, there are no more plants, i.e. food.
I could go on and on. I'm unsure why you see the survival of humanity as more important than the survival of the planet. The planet will go on long after we are gone, but right now, we cannot survive without the current atmosphere, and climate heating up just a few degrees will spread more zoonotic pandemics, kill our seas, kill our plant life, and more.
We are not more important than the planet we are on. Without the particular atmosphere of the planet we are on, we do not survive.