the private ownership of the means of production

Is there any reason you're using marxist terminology?

People have measurably shorter lifespans due to coal emissions. We aren’t doing anything about that, so why do you think property will be treated any differently?

I'm not sure why you say it's Marxist terminology. It's more or less what Wikipedia uses. Is there another definition that you'd prefer?

If you assume that capitalism requires unchecked growth harmful to the environment, it doesn't follow that some specific alternatives to capitalism are automatically healthy for the environment. The only conclusion you can draw (if the initial proposition is true) is that if there is a healthy alternative, it's not capitalism.

By the way, some people believe China is currently practicing a form of "state capitalism" (regardless of what they call it, which by the way seems to be "socialist market economy").

The basics are enough: unchecked greed is bad, regardless of the society / system of capital allocation and thought.