That's the exact opposite of what de-growthers believe (though my experience has been that most 'de-growthers' are in fact imaginary straw men since I've only met people decrying them, none truly espousing these beliefs as a strategy).
The beliefs described by the parent are basically the de facto beliefs of everyone who believes the climate crisis can be solved through some form of accelerationism.
I don't think there's even a question that the only proven method of reducing emissions and slowing climate change is to leave fossil fuels in the ground (which is by definition de-growth in practice). But there has never been any remotely serious will for actions of this nature.
What you actually have to "prove" is that you can pass/execute the laws required to "leave the fossil fuels in the ground".
All evidence thus far is that it's not happening, no one even votes for that under democracies.
So far, all evidence is that we can pass laws, create engineering, and cause behavior change that leads to reducing carbon emissions without hurting growth.
If you care about reducing carbon, the strategy of "destroy capitalism first" isn't going to happen at all, and especially not soon enough to have the impact you want.
Parent comment does not say that their preffered, nonetheless the only way to "leave the fossil fuels in the ground" is to pass laws mandating that, nor that they favor a "destroy capitalism first" approach. Leaving fossil fuels in the ground by say encouraging the adoption of another energy source like solar has thus far been a successful strategy.
The people publishing books using the term "degrowth" (Jason Hickel, etc.) do call for destroying capitalism and believe fighting for climate change isn't compatible with growth.
Green growth people (like myself) argue that we can grow without incurring the negative carbon impacts given proper policy / tech.
We very much want solar, etc. technologies that do this. Degrowth isn't the solution.
"This is what de-growthers actually believe."