So do you think the government should have this power?
Do you think a collection of random bureaucrats should have the power to throw out the First Amendment?

Those are two different questions. The answer to the latter one is clearly "no." But the answer to the first question might still be "yes," because "this power" doesn't necessarily throw out - or even conflict with - the First Amendment.

The key here is to remember that governments are not just regulators of speech. They are also "speakers" who frequently advocate viewpoints and positions within the metaphorical marketplace of ideas - even in areas where they don't actually have any regulations. To use a hopefully non-controversial example, your state government might do a bunch of PSA's trying to encourage kids to be active and exercise and play outdoors as part of an effort to combat childhood obesity. It's not illegal for kids to be overweight, or for parents to let their kids stay inside and watch TV - the state is simply trying to promote a particular viewpoint ("kids should exercise"). And they're allowed to. They might also try to persuade third parties to both agree with them and to change their own behavior to promulgate that viewpoint - staying in the same vein, they might persuade the NFL that childhood obesity is a problem and try to get the NFL to encourage childhood activity.

It's pretty clear in the case law that government is allowed to try to persuade people. They're allowed to try to get third parties to change their behavior to favor the government's preferred policies.

That's easy with positions that are not really contested. But things get complicated when the government has a very firm opinion on, but that some people disagree with. And very complicated when they're trying to persuade private parties to deplatform people on the other side of that split.

So if there's a problem with teens huffing paint, the government will go out and try to persuade teens to stop huffing paint (again, hopefully not a controversial position). They will also reach out to community organizations (say, churches and youth groups) to ask them to persuade kids that huffing paint is bad (again, not very controversial). In today's world, though, they might also reach out to Facebook and Twitter and ask them to enforce any existing policies they might have that would prohibit groups or boards that teens use to share tips and tricks on how to huff paint.

That last step is more controversial. The government is allowed to think that huffing paint is bad and is allowed to try to persuade people to take action based on that belief, just like any other speaker. But are they allowed to try to persuade private parties who own communications platforms to act on a belief that huffing paint is bad?