I wouldn't say that he didn't receive backing. Dortmund made significant investments by their standards, especially last summer in keeping their squad together while adding Bellingham. They also stood behind him while the media was sharpening their daggers after his first season collapse and they sat out his second season crisis, when it would have been an easy out to sack him before their annual meetings. He did well to stabilize the club initially, but as time went on it became more and more clear that he didn't fit Dortmund's vision of their future. So there just wasn't much of a reason left to sit this one out with him.
I probably worded this wrong, I don't mean he wasn't backed in terms of investment. I meant that while they did stand by him after earlier bad spells, as you write, they did so in a noticeably reserved manner. At least that's the impression I was getting.
I think these days it's all about ideas, giving your team a feeling of inspiration. On one hand we saw Nagelsmann salvage Hoffenheim from a desperate relegation battle and turn them into a top 4 club, despite being younger than a couple of his players. Tuchel had Dortmund on track for a record breaking season, despite only coaching Mainz before and having to take over from Klopp. We also saw Ancelotti get rejected by Bayern's squad in record time. These days you need to convince players that you can help them reach a higher level, that you have the answers to their struggles. Reputation is just bonus.
If Pochettino was German speaking sure, but the language barrier by itself could make this a disaster. And that's too big a risk in Covid times. And that's all assuming he would even want the job. The fact that they announced Terzic until season's end basically puts any idea of Pochettino or Rangnick to rest immediately anyway.
I absolutely agree about the bold part, and that's also in part what I meant with authority. The players need to be convinced that the coach knows how to set them up well and improve them (their careers), but they also need to see the proof of success in training and in games. And then, a good reputation can mean the burden of proof is not immediately there, or lower, as its connected to what the coach achieved with/for his players before.
For example, I don't think Ancelotti was rejected by our squad in record time (his first season was quite successful after all), I think the players responded to him well initially, but we let Paul Clement go in midseason and after that my guess is the standard of training got worse, what with Ancelotti putting his son in charge. And that's when our players, used to a high standard in training from Pep, started to worry. Among other things.
But yeah, it is obvious that Nagelsmann is uniquely infectious, or convincing, as both coach and motivator, and that's the aura surrounding the best managers.
Still he arguably needed, and needs, to build up a certain reputation and authority, a myth if you will, or a certain gravitas, before taking over a job at the biggest clubs. Because at those there might be forces at play that can easily be bigger than the brightness and plausibility of a talented young coach's ideas. Which is also why he probably was smart to decide against a move from Hoffenheim straight to Real, if that story is true.
Rose next summer sounds like a very good move (just wonder at what point the footballing public will catch up to the fact that BVB's bullying of Gladbach is easily as bad as the usual accusations towards Bayern's grip on the competition), he seems like an outstanding manager and a very likeable guy.
Is he a better manager than Pochettino? I think Poch's achievement at Spurs are not to be underrated, even if hiring him would present obstacles and problems. Rose certainly has not a better record than that. But you're right in that those obstacles might make it too risky an appointment for the taste of the board.