The main problem with that approach I believe is that since proving your age to the porn site involves an interaction with the government site and the porn site

The main problem with that approach I believe is that since proving your age to the porn site involves an interaction with the government site and the porn site which will occur at close to the same time. If the porn site and the government site both keep logs and someone is able to get the logs from both they might be able to match up timestamps to unmask some porn site users. A system that allows this is called "linkable". It might be possible to mitigate this by introducing delays and noise. Wait some random amount of time before giving the finishing the verification with the porn site. Have a process running on your computer that frequently generates nonces and has the government sign them. Probably better is to use a protocol that is unlinkable. There are ZKP systems that are unlinkable.