I think basically everyone should support a carbon tax. It's a really obvious solution that is both environmentally friendly and should be acceptable to free market fanatics because it is explicitly and only taxing a negative externality on the public - it's hard to imagine a more justified tax.
Combined with the increased cost effectiveness of renewables & batteries, & the new build-out of nuclear, it could plausibly speed up the clean energy transition, rather than just disincentivising building out more polluting power plants.
There are two main options for what to do with revenue from a carbon tax. The one that makes the most macroeconomic sense is to use those proceeds to fund subsidies for clean energy roll outs & grid adaptation. You are directly taxing the polluting power grid to fund the construction of a non-polluting power grid. As CO2 emitting industry (and thus carbon tax revenue) declines, we have less required spend on clean energy roll out, so the tax would balance nicely. The downside would be that a carbon tax would increase cost of living and this does nothing about that.
The other option is a disbursement. Give everyone in society a payment directly from the proceeds of the carbon tax. This would offset the regressive aspects of a carbon tax (because that tax would increase consumer costs), and would also act as a sort of auto-stimulus to stop the economy from turning down due to consumption costs increasing. The downside of this is that the clean energy transition happens slower than the above, and that there may be political instability & perverse incentives as people maybe come to rely on this payment that has to go away over the next few decades.
They're both good options. I don't know which is better and I think that's likely something individual countries will probably choose based on their situation. But we do need some sort of way to make those emitting CO2 pay for its negative externalities.