This is a really weak argument. High-quality government websites exist, most notably the British gov.uk system. All this is beside the point - the current age verification proposals are wrong, and attempt to solve the wrong problem in ways that look very much as if they are primarily intended for surveillance rather than child protection.
'Child safe' internet access by default with a very low bar to let adults enable their devices to access the public internet is the solution. The UK's mobile providers already offer this by default - all you have to do is to call your network provider and ask for the restrictions to be turned off, and you have full internet access. Any legislation to create 'child safe' filtering should mandate that vendors provide ways for adults to turn the filtering off, easily, quickly and no questions asked - and that it should be viewed as the responsible moral default for adults to have full access to the world in order to be able to exercise their responsibilities as citizens, not some sort of option for degenerates. Whether you do this device side or network side is a technical detail.
The other sane way to do this - and what we should have done from the start, instead of providing kids with full access to the adult world by default - would be to have created 'child safe' devices to provide children with filtered access. Parents could then be in charge of whether they want their children to have that access and at what age. However, that boat has sailed.