Does this site even make money?

Ok, assuming that we're treating HN like a business and that business is large enough to have significant government voting share under this scheme:

The government voting with its shares to silence those stories would potentially be actual censorship under the US constitution, and therefore illegal. The same cannot be said about private owners.

Of course, the government could opt to transition the site away from those topics entirely, but again that's something the private owners could do too.

And none of this prevents small competitors from coming in to run those stories.

Let me turn it around: Would Facebook have more or less freedom of speech with government control?

I’d assume Facebook would be appointed a partisan czar during each administration, much like many government agencies. Unlike the EPA or FCC though, controlling Facebook has huge political implications.

Right, so it is best we leave that to Mark Zuckerberg, who surely has no political agenda whatsoever.