Whether such a system could be bypassed by a VPN would depend on exactly how the age verification works and whether said government decides to ban the use of VPNs.

More importantly, I don't personally have any faith that at least the US government could properly define and build a system that is reliably and provably resistant to tracking. The government has incentives to want to know what sites a person visits, the NSA would be loathed to allow that opportunity to go unused. The government also likely doesn't have the skills or resources to do it in house, I'd expect them to outsource it at an absurd cost to a third party that would also have incentives to want to track usage data through the system.