It would be hard to dispute that the Hulk was the worst Avenger ever. As a team-mate, he was surly, uncooperative, and had questionable hygiene. Worse yet, he would later prove to be an actual enemy to the team, and cause them no end of headaches. On top of this, he was only a member for the first 2 issues of the title (and maybe again in issue 100), making him also the shortest-tenured Avenger. Yet the Hulk was a founding member, and one way or another, he has had his impact on the team, despite the brevity of his career with them.
If one examines the Hulk's interactions with the Avengers, with very few exceptions, they all revolve around him fighting the team, even when he was an actual member! In the first issue, the heroes are gathered by Loki tricking them into battling the Jade Giant. In the second issue, they are again tricked into fighting Hulk, this time by the Space Phantom, who is imitating him. By the time the deception is revealed, the Avengers have made it clear that they don't trust the Hulk. He astutely recognizes that not only is he disliked, but further association with the team may prove his undoing: “You all hate me…fear me because I’m the strongest! If I stay with you, you’ll find some way to destroy me!” Accordingly Greenskin quits the team and takes off, only to fall into cahoots with the Sub-Mariner. The twosome battle the Avengers in issue 3, and from that time on, the Hulk would become a recurring foe.
While the battles between the Hulk and the Avengers have been numerous, one early encounter stands out for producing something positive: the first meeting of the Avengers and the Fantastic Four. In Fantastic Four #25, the Hulk reads a newspaper article (yes, this Hulk could actually read) and discovers he’s been replaced in the Avengers by Captain America. He jumps to the conclusion that his pal Rick Jones has left him to become Cap’s partner. Enraged, he heads off to New York to beat the tar out of his former group. But the FF encounters him first, and the Thing makes a valiant but hopeless stand against him. When the Avengers finally show up (in the next issue), they bump heads with the FF at first. The two groups can’t seem to coordinate their efforts, and neither is willing to concede authority to the other. The Avengers make it clear that they feel it is their responsibility to defeat their erstwhile team-mate. Ultimately though, the two super-teams unite to end the Hulk's destructive New York romp.
Despite all their troubles with him, the team didn’t give up on the idea of bringing the Hulk back into the fold. When Iron Man and the other founders hand the reins of the group over to Captain America in Avengers #16, Iron Man encourages Cap to seek out the Hulk as a member. While Iron Man says that the Hulk's strength could be useful to Cap's less powerful team, there's also the sense that he might be trying to redress the past situation with the Hulk. In all of their previous encounters, Iron Man seemed to be the one Avenger who actually tried to reason with the Hulk. His sense of responsibility towards the Hulk has continued to current times, with disastrous results.
Honestly, it's unlikely that Earth's Mightiest could have kept the Hulk in line for any length of time. His inherent nature abhors order and compliance (despite his tenure with the Defenders, which was less a team than a bunch of guys hanging out at Dr. Strange's sanctum). But they seemed to have a hard time of letting go of the idea of the Hulk as an Avenger. The Avengers are still stuck on this thought as late as Incredible Hulk #128 (in 1970 – Avengers #76 was out the same month). When the team is brought in by General Thunderbolt Ross to try to capture the Hulk, Goliath (aka Hawkeye) tells Ross, “Y’see, we’ve got two reasons to hogtie that green-skinned galoot! For his own sake---and so we can maybe add some more muscle to our team!” Predictably, they wind up spending more time trading punches than words.
Although he shows up and goes peacefully with the team to Olympus in Avengers #100, Hulk’s major contribution there is sitting around listening to music and putting the moves on a terrified Enchantress. But at least he didn’t smash anyone.
By the time of the Avengers - Defenders War, Hulk’s association with his original team is pretty much completely dissolved. He recalls being an Avenger – “didn’t like it!” – but no one is advocating for him to come back.
Probably his only real contribution to the team was not a direct one. He is, after all, responsible for the creation of She-Hulk, his cousin, who has been a valuable and loyal Avenger for many years. Although she too has had her bouts of unpredictability - just ask the Vision.
It does seem like whenever the Hulk has gone completely bananas, the Avengers have been there to take him down. To be fair though, it's not as if the Avengers haven't also tried to help the Hulk. Once the true nature of his transformation was known, they worked with other heroes like Mr. Fantastic to find a way to cure him. They showed up as character witnesses when he was on trial (Incredible Hulk #153), and they even supported his being granted amnesty later on for all his past misdeeds (Incredible Hulk #278).
But as the Hulk's destructive toll grew over the years, the heroes became less patient with him, less focused on finding a way to help him than on a way to stop him. This culminated with the decision by Iron Man, Mr. Fantastic, Blackbolt, and Dr. Strange to send the Hulk to an uninhabited planet, where he would be able to live out his life in peace. Unfortunately, the Green Goliath wound up instead on a war-torn world where he was enslaved and suffered both physically and emotionally. The Hulk returned to wreak even more destruction on Earth, particularly on those four heroes.
While it’s likely that the Hulk would never have been able to truly fit in to such an organized super-team, the inability of the Avengers to somehow find a place for him among them and curb his destructive tendencies could be seen as one of their greatest failures.