I know of no evidence that the German High Command fostered or supported the NAZI's. Thy knew of and monitored the NSDAP, as they monitored all political parties. Hitler, an itinerant corporal, was recruited to attend meetings and write reports. When his reports suddenly stopped they assumed he had been discovered and killed. No one was more surprised than the military when Hitler reemerged as a party leader. Besides, even if the German Army had played a role in the rise of the NAZI party, there is no reason to think they would have done so if Germany had not lost the war.
“but the Austro-Hungarian Empire would have survived”
Probably not...
Ethnic nationalism was plaguing the Empire badly in the years leading up to WW1, and it was already “Balkanizing” when the War began, sped up by incompetent management from the top. The Empire had been in decline ever since it’s defeat by the Prussians in 1866.
As one researcher puts it, “they were living on borrowed, time, if not borrowed decades already. It was a prison of various nationalities looking for escape.”
“I know of no evidence that the German High Command fostered or supported the NAZI’s.”
In fact, much of the German High Command despised the Nazis as trash; that was why Hitler needed an SS that was personally loyal to him. The army kept it’s own salute, and independence from the Nazi Party political machine, until after the bomb plot. The German aristocracy, as well as it’s military component (officers from the Prussian aristocracy, with their traditions), saw Nazis only as a counter to communism that might restore the military to a rightful place (after the harsh conditions of Versailles.
Germany gained everything it wanted when it attacked after the October Revolution. To get the White Army into alliance they would have to cede the gains of the Brest-Litovsk Treaty back to the New Imperial Russia (because it was the heart of White territory). Which would leave them exactly where they were in 1914. The Treaty crippled Russia, With it in place it didn't matter who ruled - Romanov, Kerensky, or Lenin - Lesser Russia would be no strategic threat to Germany.
If the army was willing to do just about anything they would be willing to throw up a Hitler if that was necessary. Its view of what was a necessity might have been different but the forces behind Hitler were extremely powerful.
I simply do not believe there would have been any incentive for the Germans to turn on an ally, Russia, when they were surrounded by a hostile world. And the Austro-Hungarian empire was dead on its feet like Turkey. It may well have staggered along for a decade but had no long term future after all it was the attack of nationalism which started the war in the first place. As a breeding ground of radicalism and antisemitism further outbreaks were inevitable. You might have seen a fascist takeover in Austro-Hungary.
After all the first fascist takeover was not in Germany but in Italy one of the “victors” in WWI. These ideas were in the air and used the prevalent antisemitism to recruit.