Splendid.
Explain this to me in small words.
I did, but you didn't seem to get and you example betrays a total lack of understanding of economics, I'll explain.
A certain service that I require has costs of $500 p/a. A business will charge me this plus $100 p/a as profit. I have a choice between purchasing this service from (a) the company, which will therefore charge me $600 p/a (the cost of the service plus profit) (b) a non-profit organisation which will therefore charge me $500 p/a (the cost of the service).
It seems to me that in the second case, I am $100 p/a better off, but perhaps you will explain to me in small words how I am wrong.
Yes, you fail to consider not just why the for-profit business would want 100 dollars in profit, but why you would pay it. If making the product costs 500 why don't you just make it yourself? Probably because, even if the components for the service or products cost 500, it requires work and expertise to hammer the components together into the final product or service.
That means you aren't paying the 100 dollars in profit for nothing, you're paying for the work done. Alternatively it requires an investment (and I covered capital investments in the last post).
So if the non-profit organisation can provide a cheaper service it is because either, the workers agree to work for free at the soup kitchen (or wherever) or because someone has donated some money for which they forego the return on investment they could have gotten from this. In other words there is no efficiency gain. Only other people who are paying some of the cost of the service for you.
In that case there is of cause still a gain for you, however government is not a charity. Government workers do not work for free and if government invests money then they forego return on investment which imposes a cost on the taxpayer equivalent to paying for a private actors return on investment.
In other words of a government can provide a service better or cheaper than the private sector it is not because the government needs no profit, it's for some other reasons such as asymmetric information, monopoly or other market failure.
What you seem to be arguing is that a government wishing to make a capital investment in something-or-other might as well buy it from a businessman as make whatever-it-is itself, which may very well be true under idealised economic circumstances, but seems somewhat irrelevant to the discussion.
No, it’s not irrelevant, it’s what profit is. Profit is not some magical extra cost that a private sector tacks on because they’re greedy. Profit is simply return on investment or pay for time spent on a task. It does not in itself introduce an economic inefficiency.
In case of for example a monopoly the profit can be increased above the normal level, which is inefficient, but that’s a problem with monopoly, not with normal profit.
as well as the customer based metrics of treatment, and timely treatment, and quality of treatment received. 
I haven't done the kind of rigorous analysis on my own to both identify the correct measures on both facets of that gem. (It would be someone's full time job for some years to do it with sufficient rigor, and I'm not in the business.)
