Well, first of all, your premise of inter-war influence is contingent upon the historiography of Mongol tactics up til that point. Do you have that bibliography; a bibliography of the historiography of Mongol tactics, written up to ~1936.
That seems disputed. In the wiki article you quoted, it states:
"However, recent historians have challenged his reputation. says it is an exaggeration to credit him with a uniquely disciplined conscript army, or call his the first military state to fight a protracted war on the continent. He argues that he improved existing techniques and used them brilliantly. Richard Brzezinski says his legendary status was based on inaccurate myths created by later historians. Many of his innovations were developed by his senior staff."
Citation: Jorgensen (2001) p 229
Did I actually quote that article? No, I didn't!
Much history is disputed, this comes with the discipline. The points in the article that are disputed directly have less to do with G A's influence upon the evolution of combined arms tactics and more to do with other historical precedents; this is stated, explicitly! Citations of primary sources (and not historians interpretations thereof) would either support or undermine the assertion that he may have taken credit for contributions of his subordinates, but these are not presented. And, Hart and other historians are NOT, as you suggest, primary sources; they are researchers and interpreters thereof! Additionally, Hart was as much military-myth reproducer as he was military historian!
The stuff mentioned in the article which is credited to Gustavus was all used by the Mongols against Europe centuries before this guy came along. So the title of "innovator" can't be applied to him. And Hart is one of the historian-military theorists I mentioned in the OP.
From the article: [...] "His innovative tactical integration of infantry, cavalry, logistics and particularly his use of artillery, earned him the title of the "Father of Modern Warfare"." [...] Really, the Mongols integrated infantry into their tactics during their European campaigns? Please provide a bibliography of the historiography of Mongol infantry tactics during the invasions of Europe.
Not by WWII. The German Officer Corps (it started with the Prussians under Moltke) had already instituted a meritocratic system in the leadership, and the rest of Europe followed suit, as far as I know.
Perhaps! However, I would argue that Western meritocracy is a "qualified" meritocracy!
The arguments I've heard from Carlin, were that it was in the inter-war period, after WWI, where Mongol influence took hold in Europe. This was due in-part to the events of WWI itself, where manoeuvrability was completely taken away by the obsolescence of cavalry (due to the machine gun.) So as the experts thought about how to regain maneuverability, they eventually rediscovered the Mongol Army's achievements, which is hard to believe that it took them this long... since it was no doubt the most successful army ever (even today.)
What sources did he cite?
I suspect Mongolian contributions have been ignored by everyone mostly because they are politically incorrect... No one likes the Mongols (in the East or the West). Of course they were horrible and genocidal, but their actual military accomplishments are undeniable. I mean, it's shocking how so far ahead of their time they really were, when you study their campaigns across Eurasia. Everyone hears about the armies of Alexander, and Rome etc. But what the Mongols accomplished militarily, nothing even comes close.
99% agreement, 1% disagreement! They were not ahead of their time; they were right on time! The concept of anachronism is based on the notion of simultaneity which is wholly imagined and technically erroneous.
p.s. I highly recommend that podcast series that details all of this.
While maybe good food for thought, an inspiration to spark inquiry, I don't seek historical data from media in which neither sources are provided nor direct interrogation is possible!