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Book 1: Harry's mother's sacrifice saves Harry from Voldemort!Quirrell
Book 2: Fawkes saves Harry from the Baselisk and its venom
Book 4: Priori incantatem saves Harry from Voldemort
These were all foreshadowed in the books, though, so I'm not sure they came out of nowhere.
Book 7 is the worst for this though...off the top of my head...
The sword of Gryffindor in the pond (yeah, Snape put it there, but it happened to come at the right time)---and then again when Neville got it out of the sorting hat.
The Deluminator having powers that no one knew about before (transport, finding people, etc.)
The snitch having skin memory
People being able to use patronuses like cell phones
Wand magic---you could right an entire essay about this.
Also, there's a difference between a coincidence and DeM. DeM: when wrongs are corrected by a force that does not fit along with the story (i.e. a Greek god(dess) comnig down to solve everything, or some random king fixing the problems (r.e. Tartuffe))
There's no problem with some of these.
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The Deluminator having powers that no one knew about before (transport, finding people, etc.)
That was Dumbledore. Scrimegeour even said that it was of DD's own design.
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People being able to use patronuses like cell phones
What's the problem with that? Dumbledore invented this safe method of communication for the Order, and it's been talked about many times before.
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The sword of Gryffindor in the pond
It was said in the Prince's Tale that Hermione disclosed the information, and phineas heard it and relayed it to Snape.
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Book 1: Harry's mother's sacrifice saves Harry from Voldemort!
That's explicitly outlined in the books. It's not an unexpected force...it's magic (isn't magic what they've been doing the last 7 books?)
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Wand magic---you could right an entire essay about this.
That's not an unknown force coming to fix all the problems. Those were deep, subtle laws of Wand lore that were explicitly explained to the reader via Ollivander.
That talk with Olivander explained the inheritance of wands, so therefore it is fair game for the end of the book.
**also, I found that 'commentary' you cited to be a bunch or sarcasm from a serious pessimist. who ever wrote that has some issues.