Nothing really new here (for most economists). Just the old story re-told.
Back in high school, I figured out something neat about units. Both sides of an equation had to have the same units. Obvious now, but I thought it was neat, because it let me "cheat" on exams. If I was too lazy to solve a problem properly, I just looked at the units. "They want an answer in miles per hour, so I'm just going to divide the miles number by the hours number". It meant I didn't have to actually read the question, and it usually worked.
Some years later I discovered you could answer a lot of GMAT questions with the same trick. Decades later I stumbled upon the Wikipedia page on Dimensional analysis and discovered that real scientists had been using much more sophisticated versions of "my" trick for centuries to help them solve some serious questions. Plus, they had added the assumption that Nature doesn't really care whether we measure her in pounds or kilograms.
David Hume reasoned that the economy shouldn't really care whether we measure her in pounds or pence (OK, crowns). Milton Friedman (pdf) said the same thing, but added that it also shouldn't matter how quickly we change from measuring her in pounds to measuring her in pence. But both allowed that it did seem to matter nevertheless, for some reason, at least for a short time.
An awful lot of macroeconomics is based on the same little trick that I used on high school exams. Back then, I like to think that I was just too lazy to read and understand the exam questions, so I used my trick. Nowadays I know that the exam questions posed by the economy are just too hard for me to understand, so I, like many macroeconomists, use "my" trick to "cheat".
If you think of the structure of the economy as being described by a system of simultaneous equations, then the solution to those equations describes where the economy is right now, where it was in the past, where it will be in the future, and where it would be if something were different in those equations. You can call that solution an "equilibrium" if you like, but it may or may not have markets clearing, and may or may not be changing over time. Really, it's just a solution. And there may be more than one solution.
I don't know what those equations are, and probably couldn't solve them even if I did. But I and other macroeconomists can still use "my" trick to "cheat" and guess something about the solution.
If any equation has "dollars" in the units on one side, it must also have dollars in the units on the other side. And if you double all the variables measured in dollars, holding other variables constant, the equation should still hold true. If it doesn't, I want to know why.
There might be a reason why. If we used cowrie shells as money, for example, it wouldn't work exactly. Double the number of cowrie shells, and double all prices measured in cowrie shells. Would people still want to buy the same number of other goods as they did before? Probably not, because twice the number of cowrie shells would be twice as heavy to carry around; they would want to get rid of some by buying more other goods. Each cowrie shell also has the units "grams", and that physical dimension matters. The physical dimensions of money might also matter if we used gold, because gold has non-monetary uses where those dimensions matter. The physical dimensions of dollar bills probably matter far less, so we are usually safe to ignore them. Unless we are talking about Zimbabwe dollars, and there's not enough room in your pocket to hold them all.
The Quantity Theory of Money, and the Neutrality of Money, are merely applications of the same sort of trick I used in high school. They are not based on any particular model of the economy, but on a general property that any model of the economy should have. Even if we don't have a clue about what the equations are, and how to solve them, we know that doubling all dollar variables and holding all other variables constant should give us a second solution to the system of equations. If it doesn't, I want to know why.
Maybe, as with cowrie shells, we can't separate the physical from the monetary dimensions of money. Or maybe some variables with dollar units just can't or won't change when we change all the others, like sticky prices, or people's expectations of variables with dollar units, or old debts measured in dollars. Then we need to try to figure out how big those effects might be, and how long they will last.
We use the same trick when we draw the Long Run Aggregate Supply Curve as vertical. We might have no idea where it comes from, or if it even is a supply curve. (I don't think it is a supply curve, since in the short run you can get people to sell more than they "supply" or want to sell, which seems a bit strange). But if you put a dollar variable on the vertical axis, and a real variable on the horizontal, whatever it is you are drawing should be vertical. If it isn't, I want to know why.
The Long Run Phillips Curve, and the super-neutrality of Money, are a bit trickier. Because now we're asking whether the rate of change of all dollar variables, not the level, should make any difference to non-dollar variables.
Inflation is clearly the rate of change of a dollar variable, the price level, per unit of time. Interest rates are tricky. We think of a nominal interest rate as being a nominal variable, and a real interest rate (adjusted for inflation) as a real variable. But both have the same dimension 1/time, with units like percent per year. I think of the nominal rate as the rate at which the dollar value of what's in my savings account is increasing over time. So it's the rate of change of a dollar variable. And the real rate is the rate at which the real value is increasing over time.
One nominal rate of interest clearly can't change when the rate of change of all other dollar variables increases. The nominal interest rate on holding currency is stuck at 0%. So the real interest rate on holding currency must decline when inflation increases. At Zimbabwean levels of inflation, that probably matters a lot, for a lot of other real variables. It probably still matters, though a lot less, at lower rates of inflation.
But again, if you draw a Long Run Phillips Curve, or any curve with the rate of change of a dollar variable on the vertical axis, and some non-dollar variable on the horizontal axis, and you don't draw it vertical; I want to know why.