It is a stretch, which is why it needed to be explained.
And yes, it would kind of make him immune to dying... in cases where he could be accidentally rescued. Cases like a first year student's spell locking a door, which an investigator could easily dispel when trying to investigate.
Oh, and I guess once it was established, the other time travel scenes would have had to be written differently. Or at least clarify that "while Draco's murder plot was flimsy enough that the simplest timeline was the timeline in which it failed, Quirrel's murder plot was bulletproof enough that the simplest outcome was for it to succeed." Because authors write the rules, they can get away with a lot of nonsense. But in this kind of story, they do need to acknowledge and (try to) explain any inconsistencies.
And here's the line I was referring to:
"The earlier experiment had measured whether Transfiguring a long diamond rod into a shorter diamond rod would allow it to lift a suspended heavy weight as it contracted, i.e., could you Transfigure against tension, which you in fact could." (Chapter 28, foreshadowing the nanotube, which may or may not have been what you were talking about)
I don't mind the occasional protagonist who makes their own trouble. I agree it would be annoying if all protagonists were like that (and I agree that Harry is annoying in general), there's room in the world for stories like this.
Now that you mention it, your first example does sound like a Deus Ex Machina. Except that
the story already established that the simplest possible time loop is preferred, and it's entirely possible that if Harry hadn't gotten out to pass a note, someone would have gone back in time to investigate his death, and inadvertently caused a paradox by unlocking the door.
This wouldn't have had to be a long explanation or full-blown lecture, just enough to confirm this interpretation. But since it wasn't confirmed and there are multiple valid interpretations of the mechanics, it does come across as a bit of an "I got out of jail free" moment.
I... don't understand your second example. I think that part of the story works just fine. Harry's solution was plausible, and even foreshadowed
in chapter 28 when he used transfiguration to apply force.
It's been a while since I read it, but off the top of my head I can't recall any blatant cases of Deus ex Machina. I'd ask for concrete examples, but I don't think it would be useful. I'm sure you can provide an example, and in turn I'll point out reasons why it doesn't count as Deus ex Machina. We'd argue about how well the solution was explained, and whether enough clues were presented far enough in advance to count as valid foreshadowing, and ultimately it'll come down to opinion.
Instead, I can go ahead and answer your question. Eliezer definitely meant to teach useful lessons. Not everything Harry does is meant to be a good example (I mean, even Eliezer knows better than to write a completely perfect character), which is probably why he gets into all that trouble. But whenever a character goes into Lecture Mode while solving a problem, it's meant to be both useful and accurate.
Wait a minute, are you talking about Lecture Mode when you say "Deus ex Machina"? I can kind of see that: the situation seems hopeless and then someone (usually Harry) gives a long explanation and suddenly the problem is solved. Thing is, these lectures don't pull the solution out of nowhere. The relevant story details were established beforehand, and the lecture just puts them together. (Or at least, that was the author's intent. As I said, it comes down to opinion.)
An example I like is the Knight Capital Group trading incident. Here are the parts that I consider relevant:
KCG deployed new code to a production environment, and while I assume this code was thoroughly tested in a sandbox, one of the production servers had some legacy code ("Power Peg") that wasn't in the sandbox and therefore wasn't tested with the new code. These two pieces of code used the same flag for different purposes: the new code set the flag during routine trading, but Power Peg interpreted that flag as a signal to buy and sell ~10,000 arbitrary* stocks.
*Actually not arbitrary. What matters is that the legacy algorithm was optimized for something other than making money, so it lost money on average.
They stopped this code after 45 minutes, but by then it was too late. Power Peg had already placed millions of inadvisable orders, nearly bankrupting KCG.
Sometimes, corrigibility isn't enough.
I suppose, but even then he would have to take time to review the state of the puzzle. You would still expect him to take longer to spot complex details, and perhaps he'd examine a piece or two to refresh his memory.
But that isn't my true rejection here.
If you assume that Claude's brother "spent arbitrarily much time" beforehand, the moral of the story becomes significantly less helpful:
"If you're having trouble, spend an arbitrarily large amount of time working on the problem."
His brother's hint contained information that he couldn't have gotten by giving the hint to himself. The fact that his brother said this while passing by means that he spotted a low-hanging fruit. If his brother had spent more time looking before giving the hint, this would have indicated a fruit that was a little higher up.
This advice is worth trying, but when you give it to yourself, you can't be sure that there's low hanging fruit left. If someone else gives it to you, you know it's worth looking for, because you know there's something there to find. (The difference is that they, not you, took the time to search for it.)
Again, it's a worthwhile suggestion. I just want to point out that it boils down to "If you're having trouble, check for easier solutions," and that while you can always give this advice to yourself, it will not always help.
Harry left "a portion of his life" (not an exact quote) in Azkaban, and apparently it will remain there forever. That could be the remnant that Death would fail to destroy.
Anyway, Snape drew attention to the final line in the prophecy. It talked about two different spirits that couldn't exist in the same world, or perhaps two ingredients that cannot exist in the same cauldron. That's not Harry and Voldemort; that's Harry and Death.
I mean, Harry has already sworn to put an end to death. It's how he casts his patronus. He's a lot less sure about killing Voldemort, and would prefer not to, if given the choice.
On the other hand, MIRI hit its goal three weeks early, so the amount of support is pretty obvious.
Though I have to admit, I was going to remain silent too, and upon reflection I couldn't think of any good reasons to do so. It may not be necessary, but it couldn't hurt either. So...
I donated $700 to CFAR.
So that's how Omega got the money for box B!
Well designed traditions and protocols will contain elements that cause most competent people to not want to throw them out.