The challenge with "protect the children" is not only evildoers targeting them, but targets actively seeking things out. They'll be the first ones looking for ways to circumvent age verification.
Children should be safe online and in school or nursery.
We should try our best to protect children online in a similar way that there are age restrictions in the real world like for alcohol and movies. It won't ever be 100% but for parents who care it helps greatly.
For abuse perpetrated by an adult on children we should strive to stop 100% of cases, but unfortunately that is not possible either.
Open online spaces directed at children are more dangerous for children than the average open online space. Age verification doesn't help at all with that.
That said, I don't expect this to happen, switching is very hard for many reasons.
Historical precedent: prohibition.
Alternate future: the big websites start losing billions because people just use the internet less or not at all because it's a hassle with no return, and tax revenue drops. Then the politicians start to worry.
Even in the absence of democracy, public opinion affects politics.