Burnout doesn't usually come from work itself, but from its effects on your home life. As someone who has worked in startups, has a family, and takes on too many extracurricular activities, these three things helped me the most:
I set limits on my time spent thinking about work. I will put in a solid 40 hours (and more if my company really needs me on occasion), but I let it go after my 8-hour allotment. It will take a few days for both you and your company to get used to this. However, don't hesitate to prioritize your family commitments. For example, you might say, 'Sorry, that's all for today. I need to attend to family responsibilities.' and then firmly adhere to this boundary.
Take small amounts of time off. The effects of having time off are near immediate, but they don't last long. Instead of taking a solid month off, plan several 3-4 day weekends well in advance, and it will pay off massively.
Get your personal/outside-of-work house in order. Let your loved ones (or people/groups with whom you've made commitments) know that you're trying to mitigate your burnout and outline what you need. Whether that's some time to yourself each day or scaling back your commitments, it can help immensely.
The current approach is perceived as an attack based upon moral superiority by those who honestly believe climate change is not manmade and/or stand to lose money from addressing it.
Attacking the moral compass of those who believe they are acting morally right (or pretend to be) isn't working. We need to find common ground with these people. It may need to start with financial implications of not addressing climate change.