The point I was trying to make wasn't that dollars shouldn't be used to measure value, but that if you're going to do so, those dollars need to actually correspond to something that isn't just pieces of paper or bits in a bank computer. It's not particularly interesting for $450 dollars to change hands in a vacuum; the transaction is given meaning because it represents something about the painting that was created and sold.
Similarly, velocity of money is given meaning because of the premise that each transaction functions to optimize resource distribution (i.e. a typical trade is in the interest of both parties). If the farmer and the mechanic from the Wikipedia article were to game the system by continuing to sell the same $40 of corn to each other over and over, velocity of money would technically increase, but then it would no longer meaningfully represent the health of the economy. The validity of the measure depends on this sort of double counting being precluded or accounted for (at least, in principle it does; I have no idea how much this kind of confounding factor shows up in real life).
The biggest intended takeaway of my post wasn't even that the dollars being discussed don't correspond to anything, it's that they correspond to a bunch of different types of thing that don't sum together cleanly. That I had to make a bunch of finer economic points to get there was kind of an accident, and one that I might reframe the article to avoid if I were to rewrite it today.