What if the Luftwaffe had adopted a completely different doctrine when they began structuring.
Such a radical change is unlikely bereft of hindsight and Germany's geographical and military situation. The cult of Strategic Bombing was in many ways a natural evolution of thought that was grasped the most tightly by those nations that wished to avoid another costly ground war. For Britian, whose army traditionally was small in comparison to other continental powers, protected by water, Douchot's theories were both terrifying but also provided a potential means by which another Ypres or Somme could be avoided, hence the costly investment in 4E bombers. For the US, protected by two *oceans*, and rich enough to look for technological solutions to traditional (and costly) ground operations, 4E development made sense given the political and geographical situation. (recall that the B-17 was originally envisioned as a naval deterrent as well as SB.)
The developing Luftwaffe did contain a strategic element but it did not take root, because the Germans didn't place high stock in the pre-war notions that Strategic Bombing promised. (Recall that the Germans experimented with such operations during WWI, which while making great press ultimately achieved nothing towards their cause) Given limited resources, and facing potential land enemies on both flanks, it can also be argued that the developing Luftwaffe didn't have the "luxery" of producing both a conventional Operational airforce capable of directly assisting a traditional ground campaign as well as one that might hit distant 'strategic' targets that might eventually in the long term assist in a war. Therin lay another point. Germany wasn't planning for long wars.
There is some minor wiggle room revolving around the devotion and attention given to dive bombers and bombing at the expense of newer and bigger 2E/4E bombers but ultimately i personally don't see how the Luftwaffe could have developed any other way than it did. Their airforce was integral first to German operational victories over their opponents and then later, in the army's survival during the tougher times such as in winter Russia.
As others have already mentioned too....fully embracing a "strategic airforce" would require substantial resources that would have had to come from other areas, including the Army. One need only look at the size of the USAAF as it grew from tentative beginnings in 1942 to the monster org it became in 1944-45 (along with BC).