It's not true that no one talks about this - people talk about the demographic aspects of both fascists movements in general and in leadership recruitment, but both can be a bit specialist.

There are a couple of things going into the youth-skewed Nazi leadership, much of which it shares with other fascist movements. This is more pronounced in the Nazi case because the NSDAP was especially successful at putting movement veterans in positions of state authority.

1. Lifecourse. The life history of most leaders in fascist movements typically involved serving in the Great War, then serving as a volunteer in right-wing paramilitaries in the chaos of the early peace (both against other ethnic groups in contested border regions and, of course, against communists.) This is certainly the case for the Nazi leadership - most were Freikorps veterans, which put them in a very particular age profile.

2. Demographics of fascist movements more generally. People who joined fascist movements in almost every country had a number of demographic skews - they were more likely to be male, middle-class, involved in certain economic sectors, and young, and all of these skews are more pronounced for people who joined the movements early in their existence.

Why should fascists skew especially young? Well, there are a lot of reasons - many historians point out a "cult of youth" in fascist rhetoric, though that could have been as much an effect as a cause - but one simple reason is that fascism was a relatively new doctrine, and most people acquire their ideological commitments in young adulthood.

3. Recruitment strategies. Once in power, the NSDAP certainly couldn't rely entirely upon movement veterans, but it did concentrate more power in them than most other right-wing governments of the period, which typically were much more dependent upon the pre-existing power structures of the conservative state. Nazis had to worry about alienating industry, the military and diplomatic establishment, and the churches, but didn't really trust any of these conservative actors, either, and so relied on their own cadres. For the above reasons, these skewed young.

To add a little to /u/onomatodoxast 's answer, there were significant push-pull factors that made the NSDAP movement and its government relatively young.

Among the push factors, the NSDAP did try to position itself as a "young" political party. One of the few unchanging positions of the NSDAP was that the Weimar political establishment and its parties were out of step with the recent experience of war and defeat. Although the conservative DNVP was a recent political party culled from various conservative parties and movements, the main political parties like the Catholic Centre and SPD all had their origins in the Kaiserreich, and so too did their political leadership. Both Friedrich Ebert and Gustav Stresemann for example, were deputies to the Reichstag before the First World War actually broke out. NSDAP propaganda during the 1920s and early 1930s often emphasized the gerontocracy of the political establishment. The NSDAP emphasized that its youth meant it not only incorporated the lessons of war and defeat, but that it was also not responsible for them either. Once in power, it also actively recruited from relatively young cohorts for political loyalty.

But pushing itself as a young movement was not the only reason why young Germans flocked to the NSDAP and the Nazi agencies of the Third Reich. The Partei-Statistik of January 1933 showed that some 70% of the Party was under forty and a majority of that 70% was under thirty, so the Party was obviously doing something to draw in young Germans. Part of the answer to this data that historians have looked at is the idea that National Socialism, along with other forms of political radicalism like the KPD, offered a party platform that appealed to this generational cohort's experiences and needs better than the establishment German political parties like the SPD. Detlev Peukert in his classic The Weimar Republic terms this generational cohort the "superfluous generation" which had came of age during the war, but was too young to fight in it. Peukert asserts that this deprived these young Germans of what the wider culture saw as a right of passage. Outside of Goering and Röhm, paltry few of the NSDAP's leadership in the 1920s had been part of the pre-1914 military. While some volunteered at the start of the war, most notably Hitler, others like Himmler or Bormann were too young to see action despite being called up in the waning months of the war. The chaos of the early Weimar years such as hyperinflation further alienated this cohort where they saw dimming economic and political prospects on the horizon. The global Depression accelerated this sense of estrangement from the existing order while simultaneously priming the ground for radicalism among German youth.

One particular appeal of the NSDAP both as a political organization and as a government was that it offered an alternative and fast-tracked career path to young German men at the start of their professional careers. The Weimar civil service and bureaucracy continued on older German precedents that rewarded seniority and age. Young professionals could be brought into the system, but there was advancing up the career ladder necessitated a great deal of patience and the right senior patron. Once in power, the NSDAP offered an alternative career pathway and social networks for political advancement. Ribbentrop is an excellent example of this process. Most of his main rivals within the Foreign Ministry were significantly older than him and often joined the party after 1933. Ernst von Weizsäcker, for examples, was a decade older than von Ribbentrop and twenty years senior to von Ribbentrop's main deputy Gustav Adolf Steengracht von Moyland. One of the advantages von Ribbentrop had within the Nazi system was that he was able to play on Hitler's own parvenu stylings and portray his rivals within the Foreign Ministry as dottering old men incapable of seizing the moment. Heydrich found his membership within the SS a welcome career path after his behavior torpedoed his naval career and he was forced to start a career path from scratch. Although he had participated in the Freikorps movement in the early 1920s, Heydrich had very little prior experience with radical groups before he turned to the NSDAP. Nor was he prepared to take over the security and police functions of the RSHA as his knowledge of police procedures was largely limited to his knowledge from the Navy and his consumption of detective novels. Nonetheless, the SS proved to be a wise career move on his part.

Many of Heydrich's underlings and coworkers in the RSHA were of the same age. In contrast to Peukert, the German historian Michael Wildt does not see the RSHA men as a superfluous, but rather as an uncompromising one. Wildt's examination of the RSHA membership denotes some of the same generational cleavages as Peukert, but emphasizes the war and its aftermath formed a number of radical ideas that only needed the right contest to be triggered. The German university system, for instance, has a strong streak of völkisch nationalism and antisemitism embedded in its student culture, so that many young German men began their careers already exposed to these ideals. The war experience created a sense of disorientation as these men participated in the home front and imbibed the wartime nationalism, but were suddenly left adrift with the defeat and revolution in 1918. Wildt notes that not only did the RSHA personnel find their SS membership a wise career choice, but its ideology of völkisch racism and Führerprinzip resonated with a Weltanschauung that had been developed a decade before. Pace Hannah Arendt, Wildt notes that the RSHA men could be both technocratic desk-murderers and antisemitic ideologues; and in practice, there was not much distinction between these two poles.

The skewing to youth proved to be a dubious long-term benefit to the Third Reich. On one hand, its successful push-pull with regards to German young adults did allow it to capture the state and its organs more effectively than had it just relied upon co-option of older fellow travelers within the bureaucracy and civil service. Co-option did occur and there were a number of older men involved within the Nazi state who represented a line of continuity between German governments, but the youth wing of the NSDAP meant there were political "minders" and upstarts within the system. But this created a friction of its own within the Third Reich and the clashes between pre-1933 establishment types and the NSDAP young turks created a dysfunction within a number of agencies. Hitler's centralized power meant he was the arbiter of these disputes, which further degraded these agencies' effectiveness as rivals sought to court Führer to settle disputes. Von Ribbentrop owed his continued success in spite of the fact that nearly everyone hated him (an accomplishment within the Third Reich) because he could present options for a radical foreign policy that suited Hitler's more aggressive instincts. Hans Globke would recall in the postwar period when he was defending his involvement in drafting the Nuremberg codes that one of the constant struggles within the Third Reich government was to produce results to state initiatives quickly lest a rival present a more radical solution that Hitler would agree with. Even if these Germans, both young and old, were not part of Wildt's "uncompromising generation," they often had to act the part of an ideologue to keep their jobs and power.