Private can work fine providing there is sufficient competition.
Absolutely. Often forgotten when people cry out for laissez faire economies.
Internet access, like other utilities, is a natural monopoly
Almost. Not internet access but fixed line telecommunications. It isn't quite a natural monopoly, but so damn close as not to matter much. Internet access is further up the chain, which is the brilliance of the NBN in my opinion. They allow the competitive environment for IPTV, internet access, and anything else involving high speed and reliable telecoms.
Back on topic, I like Turnbull's idea. Of course this isn't that revolutionary and was part of the justification for the NBN, cost savings for government services. Standardisation of the communications would be fantastic. Perhaps with secure email from certified organisations as the system in Denmark mentioned by irp below. Although, I can't see how you could get spam unless it is just a standard email address which makes the whole thing pointless.