Not all your online activity, even if they kept logs it would be something like 'this site asked for age verification, we said yes'.
So they would have a list of sites, if they stored them and were allowed to store them. Which is something they can get from your ISP regardless.
It could be used for bad sure, lots of things can. In my perfect world this wouldn't exist at all like it hasn't for 30+ years. But putting the burden on private companies was always going to create other avenues for issues.