Sad moments? The film's not overflowing with them but off the top of my head: Thor's exasperated but heartfelt plea to Loki to come home, Tony grieving Coulson and declaring that they're not soldiers, Fury looking down at Coulson's trading cards and realising how he's used his friend's prized possessions, Tony calling Pepper and her being too worried to notice the phone call etc.
Like when he gets those eggs for Auntmay in the end and returns battered and bruised and they hug
Or that moment when he is sitting alone in his room after saving that kid and you could feel how much he misses his Father
Or when he listens to his Uncle's Voice mail in the end
Or when he goes after all those thugs in a rush of revenge
Or When CapStacy Dies and he asks Gwen to go away
Basically I mean the moments when you fell that Spider Man is a human aswell,he will face the same insecurities,pain or joy that we feel in contrast to a group of superheros fighting an alien force,the way it was shown,you cant relate to any of them and never feel their joy or sorrow
Definitely
Moments of humanity is what you mean? The Avengers is absolutely chock full of heart in that case. Here are some examples:
-Bruce rocking the cradle in India, thinking of the life he never had.
-Steve relentlessly punching the bag, breaking it, sighing and going at it again showing how frustrated and upset he is.
- Pepper and Coulson happily discussing Coulson's love life.
-Coulson's asking Cap to sign his trading cards/Natasha pre-empting the question painting the picture of one co-worker being overly enthusiastic about something and the other having heard far too much about said passion.
-Steve paying Fury despite never formally accepting the bet because he's a man of his word.
-Natasha silently searching for Hawkeye on the monitors.
-Thor trying to get through to his brother by appealing to their childhood.
-Steve's reaction to getting a pop culture reference, his momentary sense of pride.
-Tony and Bruce bonding.
-Natasha sitting silently, letting herself be afraid before steeling herself because she has a job to do.
-Natasha punching Hawkeye after he's recovered, blaming him for worrying her.
-Thor and Hulk sitting awkwardly on the Leviathan, unsure how to react to each other given their past encounter followed by Hulk's punch to claim dominance.
There are so many more moments like this, that list was just a few minutes of remembering. Each and every one reminds us that these people are human and feel the same things we do, they exist for more than just punching. Heck, half of them don't even serve the plot but are present just because they resonate.
Avengers wasn't a origin movie
Functionally it was, given it had to establish the team, how they reacted to each other and the world. The characters were also all re-introduced for the newcomers.