> Protecting children wasn't a serious motivation of many people who supported it?

It may be helpful to understand the context.

Ambitious politicians have one goal above all others: Getting elected. This type of law impacts that in two ways. One, they'd really like private corporations to censor the opposition for them, but forcing them to is clearly unconstitutional and also offends voters who don't like that level of cynical authoritarian opportunism to be so transparent. Two, if you can drum up a moral panic and then purport to solve the problem you hyped up, some people will vote for you.

Politicians combine these two into laws like this. "Think of the children" laws too often pass because anyone opposed to them is painted as a monster regardless of the details of what is actually in the law or how ill-conceived it is. Their hope is that it will both disguise their underlying goal from the voters and invoke "hard cases make bad law" to create a precedent that laws in that style aren't unconstitutional, so the next one can do what the really want. Or, because the law is ambiguous and it's hard to even know if you're following it, have something that allows them to threaten arbitrary social media companies with prosecution if they don't do what the administration really wants in a backroom deal.

> But do we think people who promoted it were insincere about believing it was harmful or undesirable for children to see violent depictions?

It's not that they might not believe those things about a particular situation, it's that they're insincere about that situation being their true motivation for passing the law, and disingenuous about the ability of their law to address it. The politicians who propose "think of the children" laws are usually acting in bad faith, and when the law is overbroad with no regard for collateral damage or constitutionality, there is no doubt of it.