Why don't nation-states rollout their own social network?

One important reason is the "cool" factor. Anything that's associated with the government is uncool by default.

Not to mention everything else - privacy, state surveillance, network effects.

If there's free choice, I don't see the general population ever choosing a state-funded network over private ones.

One way to do it is for the state to take significant control over an existing private network. Another way to do it for an authoritarian state to block all networks and make it mandatory for citizens to use the state-funded network.

Many reasons. Network effects, competition from private platforms, and no clear benefit. A national social network would not have the incentives to improve the user experience. Wherever a state has its own national network, it is through a private company and the outside competition had been removed in one way or another: Russia, Iran, China.

Network effects are a big problem here. People want to communicate with citizens of other countries. Maybe a federated system could work, with nation states (among other organizations) running instances of it.

China did.

It's called WeChat.

Definitely very state-controlled, yes.

Because it's a nightmare scenario when it comes to censoring or not censoring extremists.

The only thing worse than big companies controlling social networks is governments controlling them.