Amazingly, sometimes government isn't very good at doing things

We agree that we are biased upon this point but even so it is still true that at times government isn't very good at doing things. As with this between the Coastguard and the RNLI:

Lives are being put at risk at sea because of a delay in launching lifeboats after emergency calls to the coastguard, according to senior RNLI crew. They believe a shortage of experienced staff and poor operating procedures at the Maritime and Coastguard Agency (MCA) are leading to slow responses and potentially fatal decisions.

Delays and inefficiencies between two arms of the State aren't going to surprise anyone rich in years or experience of dealing with said State. However, in what will be a surprise to non-Britons, the Royal National Lifeboat Institution is a private sector, entirely charity supported, organisation. The Coastguard is government in its glory.

There is a larger point than just to point to inefficiency. Which is that of public goods. No one does negotiate their rescue from peril on the sea (although there is much private and Admiralty law on the rescuing of assets, such as ships abandoned etc) at the time they need rescuing. The network of those able to perform rescues is a public good. The normal reaction to which is that if it's a public good, non-excludable and non-rivalrous, then government must provide it.

Which isn't, as we can see here, true. The existence of a public good is certainly a time when we should think about whether government can or should be involved. But that is not to then make that leap to State provision.

In parts of the literature you will see that lighthouses are the obvious example of when government must. No one can charge for lights, therefore in the absence of the ability of private provision government must perform the task. Which is to entirely miss that the lighthouse system for the British Isles was and is (it's a little more complex today but only recently) a private not for profit organisation, Trinity House. Enter a British or Irish port and you pay dues, some for the port, some for "lights." Those who pass by get the guidance for free, those who dock pay. Whatever else we might think about it it does in fact work.

The existence of a public good is a very good time to start thinking about government involvement, entirely so. But that is not a sufficient reason for government to either be involved nor to provide the good or service. As with the herd immunity provided by general vaccination, there are different ways to achieve the goal. The NHS performs and pays for all child vaccinations, the US systems insist they must be performed privately before a child enters school. Both work in that sense of producing that herd immunity. 

As to which system is used where and when we think a large part of it is due to historical happenstance. Lifeboats and lighthouses really got organised in Britain during the near laissez faire times of the Victorian era. Thus we've a private system of provision. The NHS was the nationalisation of the previously extant medical care system (the NHS completed its first new hospital in 1963, 15 years after NHS creation) just because post-war Britain did seem to think that government should and could do everything. Thus much of how we provide these various public goods is dependent upon just what was the idea du jour of the time we started to address that problem.

Leading to the thought that we might profitably have a proper rethink of exactly how we do address these problems. Many are insisting today that the State should be doing more - all those calls for nationalisation - but we could equally, perhaps more in fact, profitably be pondering what is it that government does do but shouldn't be?