If there are no significant emissions to tax, then fixing environmental damage done by past generations sounds like something for governments to pick up with public money. We also spend public money on forest management, wildlife bridges to create reasonably sized habitats, etc.

It's also a while out. We have bigger issues today than "who's gonna suck back out some CO2 once we cut emissions sufficiently". We'll get to that in due time.

For what it's worth, if you have unavoidable emissions today that you want to compensate, and want to also fund the people working on CO2 capturing technology, see https://climeworks.shop for a pay-what-you-want option. I don't really believe in planting trees (which is way cheaper per ton of CO2 it claims to absolve you of), and these machines apparently emit "only" 10kg for every 100kg of CO2 captured (90% effective sounds reasonable), and it's accounted for who claimed which share of the machine's captured volume so it sounds directly effective to me. It's just that the scale is peanuts and will remain so until public bodies and companies start to run out of emission reduction options and decide to use it - but at that point we better have this tech geared up.