And worst of all, Peter turning down the chance to become an Avenger at the end of the film as if anybody - let alone a high school boy - would pass up that opportunity
That's not a problem-
the problem. The problem is the inconsistency between what is happening on screen and what is happening in the story.
Peter wants to be in the Avengers. But Peter can't be in the Avengers because he's a dumb kid and no one (least of all Tony Stark) takes him seriously.
Peter finds out about Michael Keaton and where he will be (on a motherf***ing boat) and when that "will be" will be "now" and yadda yadda. He tells Stark but Stark is kind of a jerk and blows him off (notice how
1) Stark doesn't bother to clearly explain what will happen, specifically that he will call the FBI and let them handle it, and
2) At no point does Stark take direct responsibility for his lack of communication exacerbating/causing this entire fiasco in the first place) in such a manner that Peter thinks he has to go to the boat anyway after Stark specifically told him to not go to the boat. Things happen, and the boat almost sinks, people almost die, Michael Keaton gets away, etc etc.
To whit, Peter disobeys Stark, and makes the situation worse, putting innocent people at risk for little reason. Peter is "punished" by having his super suit taken away and having his childish dream shattered.
So what does Peter do in the 3rd act?
The same damn thing! He disobeys Stark again and crashes a f***ing cargo plane into New York City. Why? To save a couple crates worth of Stark stuff!
This is the dissonance. Stark is furious when Peter disobeys him, gets personally involved and
nearly sinks a boat
in the second act: but
in the third act is
totally fine, in fact,
exuberant, when Peter disobeys him, gets personally involved and CRASHES A f***ING CARGO PLANE INTO NEW YORK CITY. One which was loaded with A METRIC TON (actually several) OF VALUABLE TECHNOLOGY THAT YOU WERE PERSONALLY INVESTED IN.
Why is Stark not trying to deep six Peter to Antarctica? Why is he happy? Was the plane and the technology within insured for ten times the actual worth? Was this entire film about a Tony Stark insurance scam?***
What has changed between the second act and the conclusion that Stark wants to give Peter what he wants? Nothing. Yet the film acts like everything has changed, even though nothing on screen is indicative of that change.
That's the problem. Nothing changes from the 2nd to 3rd act/conclusion, yet Stark goes from wanting to bite Peter's head off to giving him exactly what he wants for no discernible reason.
You find the same dissonance with Peter's desire to be in the Avengers. Peter goes after Michael Keaton, ditching his prom date/crush/unhealthy childish fixation to do it. He is willing to put his friend's father in jail over this, something the film either didn't notice or banked the audience wouldn't notice because they gave Peter, in total sincerity, the line; "How could you do this to her?". Uh, excuse me, Pete, who is it that is trying to put Michael Keaton in jail? Get outta here.
Aside: Why was The Shocker at the prom? Someone probably already pointed this out, IDK, but here is my answer: originally Michael Keaton was going to chaperone the prom, which is why he was recalcitrant to steal the Stark stuff (since that was on the same date as the prom, after all). But he changed his mind, and got one of his underlings to fill in. Keaton found out literally 2 minutes before Peter encounters The Shocker that Peter is Spider Man and will probably run out of the prom- so he was already there. That underling may also have been his brother-in-law (they seemed to be a bit closer than the other members of the work force, and I think Keaton mentions in the prologue of getting The second Shocker that job)
Or they just wanted to resolve that plotline in the film and lampshaded ("Why are you here?!?" "I guess you'll never know") that there is literally no reason for The Shocker to be there.
Anyway....
So at that point, Peter wants to be in the Avengers. Alright. So what, in that battle between Keaton and Pete, changes Peter's opinion on wanting to be in the Avengers.
Again, nothing.
The closest I could come up with is that Peter is traumatized by the battle with Keaton and just wants to be a neighborhood vigilante, going after pickpockets and bank robbers- but nothing in the closing scene indicates this (which IMO would be a clever parallel between Stark and Peter). It presents it as if Peter has come to some deep understanding about how you're only a teenager once, or something, I don't remember. But at what point in the final battle did he specifically learn that lesson? Remember, he ditched prom with his crush to put her father in prison: why is he suddenly now like, "Actually, I just want to be a teenager!". He was begging Stark to let him join the Avengers; was persistently calling them night and day; was devastated when Stark closed that avenue; and was willing to ditch prom for a chance to redeem himself. Three times in the film, Peter ditches high school fun/ the girl he likes to go be Spider Man, yet at the end of the film he decides high school > Avengers, why?
It's the exact same problem. What happens in the story is unsupported by what happens on screen. There is no reason for Peter's change beyond the need to end the story as if there were some actual change. It's not that it's unrealistic, it's that the story doesn't support the conclusion.
***= "He was happy Peter stopped the Vulture." Stopped him from doing what, exactly? All Peter has to do is call Tony Stark or Happy once the flight is over and say, "Hey, that crate you're missing, that's not an accounting error, Michael Keaton stole it, he lives at 1234 Birdman Avenue, in a nice, new age house, if you hurry you can probably still catch him and get your stuff back!". The consistent third act would be something like: Peter learns his lesson from the second act, calls Stark and then goes and enjoys himself, I mean really enjoys himself, at high school prom. Stark/the FBI assemble at Keaton's house, but Keaton escapes. His empire reduced to dust, he resolves to destroy Peter Parker- so the big final battle is Keaton trying to kill Peter for ruining his life. This also gives payoff from the earlier choices when Parker chose Spider Man > High School. It gives him/the audience a sense of what is at stake when he chooses between Spider Man and High School. Parker finally picks High School and realizes, hey, you know, maybe I am missing out on something. You can't have Parker consistently choose Spider Man, and then suddenly realize High School is better without showing Parker even choosing High School at any point of the film. 
