If anything I think your "many other networks" example reinforces the point. The private sector left to its own devices came up with tons of incompatible small networks all vying to be the juggernaut.
The government, when faced with much larger problems than growing market share, instead invested considerable resources into solving a problem in a way that could handle the needs of a large national government, and then later invested more resources into making that internet a public good.
Even if the government hadn't developed its own internet protocols (and standardized on a notional private sector one), the government would likely still have been instrumental in expanding the use of some other standard. E.g. Eisenhower didn't invent the road, but his Administration sure did ensure a lot of them got built.
And they also built gateways so they could interface with each other. The pressure to connect them together was enormous.
For another example, Unicode isn't a government standard, and has replaced the various earlier competing protocols.
http://en.wikipedia.org/wiki/Unicode#History
The idea that the private sector won't standardize has plenty of counterexamples. Ethernet, for another.
Just like if the Wright Bros had not invented the airplane in 1903, someone else would have probably within 5 years. We'd still be flying today.
I suspect that the number of inventions that would not inevitably have happened once the preconditions were in place is very small.
After all, as I mentioned previously, having worked extensively with computers before the internet, the first thing someone with two computers tries to do is hook them together.
Hell, I even did my own hardware and software to transfer files from my LSI-11 to a PC over a wire. The government couldn't have stopped an internet workalike from forming if it tried.