lawsuits for what? AFAIK you have no right to privacy in this sense
It's constitutional case law that there's an implicit right to privacy in the constitution. I don't see a law that you must wear a band with your birthday out in public passing muster based on that. I don't see why existing in cyberspace changes the inherent privacy question, and in fact makes it more meaningful given ease of automation.
Constitutional scholar here! I mean, yes, that's true in a very general sense, but no court has held that age verification to gain access to a service, or even a device, is unlawful in practice.