A carbon tax that doesn't raise cost much is not going to have much effect.
Though that depends on how you interpret "much." If you double energy cost, upper middle class folks won't change their thermostat, but work class folks won't be able to afford to turn on their AC.
Most plans I've seen that involve an energy tax involve moving that money back to the working class via reduced taxes or rebates.
We're already doing that. But it's not close to being enough. Emissions need to be halved by 2030 to keep to 1.5C warming: https://www.reuters.com/article/us-climate-change-emissions/.... The recent economic shut-down from Covid-19 cut U.S. emissions by just 1/3 for one week in April: https://www.usatoday.com/story/news/health/2020/05/19/corona....
I think that "reducing standards of living by force" is an extremist position and it's unhelpful to even bring it up.
Switching electricity generation and transport towards renewables can have large impacts on CO2 emissions with very low impact on standards of living. Carbon taxes would raise some bills but the effect is much less than (for example) stopping air conditioning.