Just look at Europe and the effect of the carbon tax there. I don't have links (sorry), but there are lots of studies "proving" that that makes for a more efficient and optimized market aligned with the desired outcome.
Because carbon emissions do the same amount damage whether it's the result of bitcoin mining or running a hospital.
Carbon taxes increase costs for everything.
Everything that produces carbon, which is exactly the point.
Should we increase the cost of say, hospitals
To the extent that they are significant polluters, yes. Trying to micromanage "good" and "bad" emissions just encourages exploiting loopholes. (e.g. https://www.washingtonpost.com/blogs/ezra-klein/post/cafe-lo...)
You can try to play a game of whack a mole by taxing and regulating individual activities hoping to reduce carbon as a second order effect, or you can just target the root problem.
There's will always be another wasteful use of carbon intensive energy so long as carbon intensive energy remains cheap.
The point of the tax is to make the carbon price correct so as to incentivize correct usage across all domains.