A few things to note before starting:
Dan Carlin is a fan of history, not a historian. He says as much, and his podcasts are best enjoyed when one keeps that in mind.
The historiography of the German-Soviet part of World War II has undergone some serious changes in the last few years.
So with that being said, let's start by examining where the ideas of "the Nazis got in the way of the army" came from.
Comprehensive books like Alan Clark's "Barbarossa" or William Shirer's "Rise and Fall of the Third Reich" (a journalist, not historian, mind you) draw heavily on German sources. Some of those memoirs, like Heinz Guderian's and Erich von Mainstein's were written after the war, and do much to argue that they (the generals) would have won the war if not for meddlers (the politicians, specifically Hitler). While these men took copious notes and their insights are helpful, they are generals writing to preserve their legacy in the face of military defeat, and only writing from their perspective.
Bias aside, Clark and Shirer do make the argument that high ranking officials competed with each other and created almost fiefdoms in ways that focused more on their personal gain in lieu of the military effort. Clark is pretty clear that administration of the conquered territories turned populations that should have been hostile to the Soviets against Germany, and that manpower was wasted on subjugation and occupation.
Some newer military historians are pushing back on the influence of Hitler himself, and of the high ranking Nazis. Not only do they point out the problems of having generals assigning blame as a core of sources, but now that they have access to the Soviet archives, they see more of the picture than was available in the past.
David M. Glantz and Jonathan House have written incredibly detailed write-ups of the Eastern Front using sources from both sides. They push back and even point out that politically, Hitler was often right about how other nations would react over protests from his generals. It was only as the war dragged on that Germany was in too deep and famous miscalculations start popping up (shifting focus to Stalingrad, splitting Army Group South into A and B, changing the table of equipment for Panzer Divisions, and so on). Glantz even argues that Hitler made the right call in having the Wehrmacht hold their ground during the Soviet offensives in the Winter of 1941-1942. Both also assign more decisions to the military staff.
They have a great series of lectures that rethink common German claims for defeat on youtube and they are worth checking out if you want to see how the historiography is changing after Soviet archives:
https://www.youtube.com/watch?v=I98P1AxQRUM - House addressing the "three alibis" for German defeat.
https://www.youtube.com/watch?v=7Clz27nghIg&t=17s - Glantz addressing "Myths and Realities."
So, I know that it doesn't answer your question definitively other than to say "it's complicated," but the truth is that there is still a lot of work to do on the German-Soviet side of the conflict.
If you trust primary sources from generals, then Hitler and Nazi meddling was a root cause of defeat.
If you trust secondary sources from the 1950's and 1960's, then at least the petty infighting and factionalism in the Nazi government reduced efficiency and manpower available for the war.
Current historians are placing less emphasis on German accounts and sources explaining the end of the war, and more on Soviet sources and reasons for results of the conflict.
If you are curious about the role of ideology on individual soldiers, I haven't read any analysis into the military impact myself. Omer Bartov's "Barbarization of the Eastern Front" is more about looking at the role of ideology and indoctrination in explaining and exploring the extent of war crimes by regular Infanterie and Panzer divisions.
Chris Browning's "Ordinary Men" looks at Reserve Police Battalions and how and to what extent regular men took part in the Holocaust.
Sources and works discussed: Primary Sources: Heinz Guderian "Panzer Leader" 1952. Erich von Mainstein "Lost Victories" 1955.
Secondary Sources: William R. Shirer "Rise and Fall of the Third Reich" 1960. Alan Clark "Barbarossa: The Russian-German Conflict, 1941-1945" 1965. David M. Glantz, Jonathan M. House "Stalingrad, volumes 1-4" 2009-2014
Recommended Lectures: Jonathan House "The Three Alibis" https://www.youtube.com/watch?v=I98P1AxQRUM
David Glantz "The Soviet-German War, 1941-1945: Myths and Realities" https://www.youtube.com/watch?v=7Clz27nghIg&t=17s
I thought Carlin’s argument was that the kaiser’s army was better than the wermacht because it was “deeper” and more fundamentally sound. The wermacht was “flashy” and it had great tanks and planes and it used them to strike fast and land some early knockout punches, but it lacked the depth and staying power of the Kaiser’s army. Am I mistaken?
While I am not aware of Carlin's exact argument here, the contention that the Wehrmacht was fundamentally different from the imperial army of WWI is not a very persuasive one.
For one thing, the overwhelming bulk of the Heer and Kriegsmarine's command officers were veterans of the old imperial army. The upper echelons of OKW and OKH were almost exclusively composed of veterans of the first war and the Weimar Reichswehr. The interwar period saw no real break in these men's service records. While they were junior and mid-level officers under the Kaiser, they were not outsiders to its military system. Jodl for instance was a decorated wartime officer and became a Major in the Reichswehr's Truppenamt, a position that was more important than his mid-level status implies as the Truppenamt was the main instrument for planning for rearmament and post-Versailles expansion. Other officers such as Rommel or Hoth remained within the peacetime army. Even a National Socialist ideologue during the war like Ferdinand Schörner still had a personal and institutional linkage to the old imperial army.
Hitler's ascent certainly did inject a certain stream of politicized ideology into the newly-christened Wehrmacht, but this can be overstated. 1933 did not mark much of a caesura in terms of staff training, the formation of doctrine, or other aspects of planning at the upper echelons. Hitler generally did not interfere in the running of military during the 1930s. He in fact needed the support of the army in the early years of the dictatorship. The relative strength of the army as an internal power bloc ensured that it was able to have its way in forbidding military personnel from being active members of political parties, including the NSDAP. Hitler's main influence between 1933 and 1938 was giving the military too much of what it wanted. The Reichwehr's Truppenamt had dreamed of a post-Versailles expansion, but Hitler's rearmament drive exceeded even the wildest plans of the Weimar period. This meant some of the Wehrmacht's junior officers were less experienced or habituated into the military than was comfortable for some Reichswehr veterans.
This does not mean that the military managed to preserve its autonomy vis-à-vis the state. The Blomberg-Fritsch affair, which saw the cashiering of War Minister Werner von Blomberg and his chief lieutenant Werner von Fritsch on semi-trumped up charges provided by the SS, did see Hitler assuming the equivalent position of War Minister. The affair also saw the advancement of more servile replacements to Blomberg like Wilhelm Keitel or Jodl. But while these men were loyal to Hitler, they were not outsiders to the military and they had relatively little direct engagement with National Socialism. Other old-line officers became estranged from the regime, most notably Ludwig Beck who resigned as chief of the General Staff in 1938 over Hitler's increasingly aggressive foreign policy.
But such civil-military tensions were not uncommon in the twentieth century and officers like Beck were outliers within the Wehrmacht. And even men like Beck or von Blomberg had proven to be allies of the regime's rearmament drive until it became too risky. Hitler's move to place himself in political ascendancy over the military did not have a violent component like Stalin's purges; twelve German generals were removed, but received generous pensions and allowed to retain their military ranks and titles. Postwar German memoirists would contend that 1938 was a turning point in civil-military affairs, but this is an exaggeration. Army leaders still toyed with the idea of a coup if Hitler's foreign policy sailed into disaster, but many of them were ideologically in sync with Hitler's war of aggression. A number of officers in Poland were upset over the SS's role in security operations and managed to restrict their access in the upcoming invasion of Western Europe, but this was much more about turf wars than disagreement over the SS's brutal methods of pacification and cowing the racial enemies of National Socialism. The rampages of the Einsatzgruppen in Poland caused a degree of consternation in the military high command. On 18 September, members of the Waffen-SS unit Leibstandarte Adolf Hitler executed fifty Jewish prisoners. Although this formation was under the command of the 10th Army, its Heer commander von Reichenau did not give the order for the mass shooting, and contracted is superior for clarification, von Rundstedt who in turn telephoned Hitler directly. Hitler informed von Rundstedt that these actions fell under Himmler's responsibilities and not the military's. Hitler had authorized the SS to take command of issues of security and pacification in the wake of the German invasion, a task that the SS heads Himmler and Heydrich were quite willing to accomplish. Although there was some discomfort among Heer generals about the morality of these actions, most notably Colonel-General Johannes Blaskowitz who filed a formal protest against the SS-led Einsatzgruppen's actions in Poland, most of the dislike for the SS's methods stemmed from the fact that they often fostered a spirit of indiscipline among Heer troops brought in to supplement the strength of these security detachments. There was little to no sympathy among the upper-echelon Heer officers for the victims of the SS's actions. Even Blaskowitz at a February 1940 OKH meeting framed his opposition to indiscriminate murder as something whose main victim was its perpetrators, because "inordinate brutalization and moral depravity that would very quickly spread like a disease through worthy German human stock." It was also OKH that began planning for an invasion of the USSR before OKW in early 1940.
The narrative of a competent Wehrmacht hamstrung by a politically-incompetent Hitler was largely a construct of the postwar period. Much of the planning for Barbarossa was done by the Heer and Hitler's occasionally interfered with its execution, but not drastically so. Hitler did make critical decisions like the move towards Kiev or Moscow, yet he was not a voice in the wilderness in these moves. The stand-fast order in the winter of 1941 was the right call given the situation. But focusing on Hitler's mistakes or errors in Barbarossa misses the forest for the trees. The real crux of the matter was that Barbarossa could not knock out the Soviets from the war. The Heer's planning for the invasion was predicated on the idea that victories would lead to Soviet collapse. The fact that the USSR's recuperative powers exceeded that of the even the wildest estimations of OKH meant that Germany had very few decent alternatives when Barbarossa stalled.
Much the same could be said of both Blau, Zitadelle, and the defensive battles of 1944/45. Hitler made some bad decisions in this period, but his decisions were not done in a vacuum. The increasing reliance upon keeping Soviet territory and various Festung positions doomed many German troops. Yet as Robert Citino in The Wehrmacht's Last Stand convincingly argued there was some merit to this refusal to give ground:
Hitler was not a military genius, as he proved over and over again in this war, but he did know a few things that his generals seemed to ignore. The more ground he gave in the sector of Army Group North, the sooner Finland would treat with the Allies and leave the war. Likewise, the closer the southern front came to the border of Romania, the more likely that Romania would leave the war. Indeed, events would bear him out on these very points in the course of 1944. The complaints from Army Group South commander Manstein that Hitler was allowing politics to trump military necessity, like the rest of Manstein’s memoirs (and those of the other generals), seem out of place set against the military tradition of Clausewitz and the notion that war is “merely the continuation of politics by other means.” Like all the German field commanders, Manstein wished to fight a war without politics, as that “independent thing” that Clausewitz warned against. He apparently didn’t realize (or perhaps didn’t care) that a Germany shorn of its Axis allies would have been utterly incapable of continuing the war at all. The Finns, Romanians, and Hungarians were contributing a considerable percentage of the manpower on the Eastern Front, and holding any sort of defensive line without them was impossible.
Germany's situation was dire after 1943 and the Wehrmacht's often quite-willing participation in the savagery of German invasion and occupation meant the question of a political accommodation with its enemies was impossible.