That's sort of untrue; the history of radio and television in the US has a significant amount of moderation that did not run afoul of the first amendment. See for the current day: https://www.fcc.gov/consumers/guides/fcc-and-speech

Aside from that, you're highlighting a US-specific dynamic. Other nations have different legal standards on speech; e.g. Britain has quite harsh libel laws. My assumption is that certainly some nations (not all) would be in a decent position to offer a well-moderated platform that is in line with their legal standards.

Other governments exist. And none of them seem to have done this either. But the "why" is lacking. As is the capability, frankly.

I mean it already is to some; some people are pushing for social media to be reclassified as a utility, and some people are banging the drums about their rights being infringed because they can't say what they want on social media, citing free speech.

Twitter is going through that right now; it does not have free speech despite being led by a self-proclaimed free speech absolutionist. Truth Social was supposedly to escape Twitter and co's censorship, but it's one of the least tolerant platforms out there. It goes on.

Why is there no Government-Built Social Media outside of the US?

There was a plot point in Madam Secretary about how the FCC is underutilized for regulating things like social media or deepfake videos on Youtube. Obviously, that's a fictional drama where they can hand-wave solutions as needed, but I do feel like in the US, federal regulators are being underutilized, or overall don't have the teeth I would expect.

I could be misinformed, but couldn't a contractor or independent company just partner with government agencies (local, state, federal) and thus do away with the first amendment infringement issue?

IIRC, this is how some tech is exempt from FOIA requests (because the tech is "private" even though the end user is the DoD).

This does seem to be a major concern