Let me be more explicit on the points where I (very respectfully) disagree with your presentation.
1) You present Say’s Law as being essentially an accounting tautology, with total supply equaling total demand by definition. Kates shows pretty convincingly that this is not at all how people thought of the Law of Markets until the mid 20th century – this formulation was introduced by Lange in 1942, and subsequently christened “Say’s Identity” by Becker and Baumol in their landmark paper. Unfortunately, this essay exerted such influence that it has been more or less taken for granted, throughout all subsequent discussion within the economics profession, that this is what Say et al were getting at.
So, what did they mean by the Law of Markets? It was a logical insight into how economies work: demand in general can never fall short of supply in general. The reasoning essentially goes something like this: it is impossible to (effectively) demand Good Y without first supplying Good X, and it is impossible to supply Good X without intending to demand some Good Y (unless the producer of X intends to consume it himself, in which case there is no problem in the first place). Therefore, “aggregate supply” must equal “aggregate demand”, and any apparent imbalance between them must be due an error on the part of the suppliers of X or Y, failing to correctly anticipate the market rate of exchange for the product in question. As Ricardo succinctly put it: “Men err in their productions, there is no deficiency of demand.”
Note that there is nothing special about money in this reasoning – it holds equally valid whether or not one of the goods considered happens to be money. So in addition to being more historically accurate, this formulation is a much more powerful argument – it is not subject to any of the barter vs. monetary qualifications that are often included in modern discussions of Say’s Law, even by its would-be defenders.
2) You cite J.S. Mill as being one of the first to introduce the barter/money distinction. Admittedly, I will not be particularly surprised if you can show me where Mill actually said something like this, as it seems that you can find Mill saying just about anything if you look hard enough. But my suspicion is certainly that this idea also comes from Becker/Baumol, who used Mill’s monetary explanation of the business cycle in his “Unsettled Questions on Political Economy” in order to shoehorn the discussion of the Law of Markets contained in that essay into their own “Say’s Identity” framework. I think Kates is correct to be skeptical of this interpretation. Why would Mill bother to make such a big deal over the Law of Markets in the essay, only to then turn around say “well, it doesn’t really hold when you try to apply it to a monetary economy, and that’s why we have recessions”? Isn’t it far more likely that Mill actually meant something along the lines of what I suggested above, and that Becker/Baumol merely misapprehended it?
3) A minor quibble here, but your analogy of a country being able to pay for its imports with financial assets, rather than exports, is beside the point when it comes to Say’s Law. It is only meant to apply to an economy in the aggregate, as an individual can similarly borrow, rather than produce, in order to demand a good in exchange.