Sure, most countries have such systems, multiple different ones in fact, not all necessarily using the internet. They're usually used for emergencies only though.

Redundancy is key if you're warding against foreign interference, nuclear events and the like. But at the end of the day if you're a free democracy you let people use whatever they want for the day to day stuff and you end up having to use those platforms to reach your citizens as well.

My country tried doing its own thing a couple decades ago. It's called the minitel. Look it up if you haven't heard of it, it's pretty neat :)

Do state media platforms count? FDR had his radio broadcasts, the UK has the BBC, various countries have emergency broadcast systems. In any case, it's certainly not something a government would find difficult if they found a need, it's probably just too much more convenient to have media companies handle most of the time.

What I think more common is that head of the state is not a priori directly broadcasting through account on private page. What is quite common is public broadcaster.

How long before Trump starts sending emergency alert texts?