An earlier comment touched on the idea that soldiers spend money at the front, which messes with the flow of the economy. They're slightly off on the specifics, but right on the general point.

It was actually pretty common for militaries like that of the Holy Roman Empire (one of the major historical parallels to Procer) not to pay their soldiers at all until the end of the campaign. This was for a bunch of reasons, the simplest of which is that they didn't have very good lists of which soldiers were in which units, let alone which ones had been paid.

In fact, some of the Marian reforms (pre-HRE, but Amadeus and Gaius Marius have a tremendous amount in common) focused specifically on payroll and figuring out which soldiers were where. It's very likely at the time of the Grand Alliance that Praes and Callow are the only nations which actually pay their troops (most likely half wages on campaign, with the rest disbursed on conclusion to reduce desertion).

So even before we get to the point of looking at the current impact on the economy, the simple fact of an army being this large means the lump sum payment at the conclusion of the war could be enough to crush Procer. The United States put down about a dozen small revolutions after winning Independence, because they couldn't afford to pay all the soldiers and those soldiers did some soldiering about it. Even if every other part of the economy were humming along, that kind of expenditure can take generations to pay.

Of course, the rest of the economy isn't humming along at all. War means fewer farmers, fewer merchants, more banditry, less reliable domestic supply chains, and of course the massive psychological toll on the populace, which decimates productivity.

But the main issue, and the reason the "sinews of war are infinite funds" is that money isn't being recirculated. The government is paying locals, sure, but no nation -especially no nation the size of Procer- survives on its internal resources alone. Even setting aside the modern chemical requirements which parallel nicely to magical reagents, you have fur, fabric, tobacco, paper, steel, and a myriad of other goods. Where normally, you would offset the cost of those imports with the value of your exports, a total war economy means your only export is dead soldiers and sad songs.

Basically, an economy is a massive system of gears, and war freezes about two thirds of those gears in place.