All of the Avengers were evaluated - it was only Tony who failed the test
Some prop-Tony folks get very upset about Tony failing Shield's evaluation in Iron Man 2. They claim it was unfair because Tony was upset (which is a theme with them) and that none of the other Avengers faced the same tests. It's appropriate to test Tony while he is under stress because being an Avenger is stressful. He failed for the reasons Natasha outlined (ego, self-centeredness, arrogance), which Pepper confirms in Avengers 1. And I think the others were evaluated, just not on screen.
Note: Set in an AU where there's more time between Thor 1, Captain America 1, and Avengers 1.
***
Tony stormed into Nick Fury's office and said, "I demand a recount."
Nick Fury looked up from his file and asked, "What are you talking about?"
Tony folded his arms and said, "That report Rushman--Romanov did on me. Saying Tony Stark was Not Recommended even if Iron Man was."
Nick leaned back in his chair and said, "What about it?"
Tony replied, "I heard you actually put a band together. Did any of THEM go through your secret psych eval?"
Nick nodded and said, "Yeah, they did."
Tony's jaw dropped. He said, "That's not possible!"
Nick snorted and said, "Why, because you think nobody could pass?"
He continued, "I've known Barton for years. When he brought Romanov in I was ready to kick her right back out. But she's proved herself over and over again."
Tony sulked and said, "Yeah, them, sure. I'll give you that. But the rest of your new crew--no way."
Nick shrugged and said, "You can believe me or not. Thor proved himself to Coulson when he was willing to die to save the folks in New Mexico. Banner has stayed under the radar and out of trouble for years and is willing to work with us as a scientist, not a superhero. Steve Rogers was one of the most successful unit commanders of the Second World War, and inspired an entire nation."
Nick sat forward and said, "And you know how reliable Colonel Rhodes is."
Tony's jaw dropped again. He said, "You picked--but I--over ME?"
Nick said, "Why the hell wouldn't I pick a man who gets the job done without needlessly endangering civilians? Not to mention someone who actually understands what it means to be part of a team."
Nick stood up, took Tony's arm, and moved him to the door. Nick said, "Listen, you can always pull your head out of your ass and change your ways. Or we could have a global catastrophe that leaves us no choice but to call you in. But otherwise...Tony Stark is still not recommended."