It's a shame (non-military segments of) governments don't do this kind of projects. Instead we're spending tons of money on rusting tanks, flying killing machines and what not.
Would you really trust government-controlled internet provider? I'm not usually one to spout conspiracy theories, but even I'd be unable to prevent doubts about the content coming through it.
I don't really think there's a difference between what we have right now and a government-controlled provider. The major backbones are already working hand-in-hand with the government, major news sources are working with the government, social news sites have government-published stories. There's no way to validate that anything isn't from the government on the Internet.
The only difference would be in perception, not in reality.
I don't know that I agree with your first paragraph, but validation of what's from the government isn't the only problem. In the internet age, freedom of speech effectively requires internet access. If the government can boot you off the internet or ban your content, that's a major blow to freedom of speech.
Let's pretend the US ended all military spending. Would you think that wars and defense would just become irrelevant? A strong military prevents wars. Would Taiwan be free of it weren't for all those "flying killing machines?" Would South Korea or Japan even exist if it weren't for the military strength of the US? The main reason we didn't go to war in Europe against the Soviets was because of all of those 'rusting tanks.' Communism didn't fall because of luck; the US outspent and out produced the Soviets and bankrupted them. Threats to freedom still exist and they aren't stopped without diplomacy backed by the threat of force.
Most of the most significant innovations of modern history came about because of free-market capitalism, not because of governments -- with very few exceptions.
The role of government isn't to innovate -- it's to maintain an environment that makes innovation possible.
I would say that many significant innovations have been driven by the military applications. Rocket technology and drones are 2 that come immediately to mind but I am sure there are many others. Not defending the military industrial complex though.