I wish Tim Roth's Abomination had survived the Incredible Hulk movie. Maybe he did, but I've always assumed Hulk broke his neck or strangled him with that chain at the end. Regardless, he hasn't made a return appearance since. Mostly I wanted him to encounter Captain America.
Ross initially pumped Blonsky full of an incomplete knockoff version of the Super-Soldier Serum, but beyond that, Blonsky is what Erskine was afraid of, what would have been the result if Colonel Phillips had his way and somehow got Gilmore Hodge chosen as the subject. Erskine feared giving more power to someone accustomed to it, that they wouldn't have any respect for what they could do with it. They'd regard everyone else as lesser, weaker beings to abuse as they saw fit. Here's Blonsky, used to being the baddest, toughest guy on the block, but fearing not only the loss of some of that prowess due to age, but also to being completely outclassed by this big, green monster.
Him taking the knockoff serum is understandable, since at the time he was going to be confined to a hospital bed for the rest of his likely short, certainly painful existence. But once he had it, saw what it could do for him, and saw that it still wasn't enough to stop Hulk, he wanted more. And once he had that, the world was a playground, something to smash and destroy as he saw fit until he could find a proper challenge.
Blonsky isn't Johann Schmidt. He doesn't have larger aspirations of ruling the world, any more than Steve Rogers does. Still, he might take orders from anyone, or attack anyone, simply for the chance to test his power, prove his superiority. A super-soldier with no interest in serving, or helping anyone but himself*. The Abomination is trashing a city, Captain America, or whatever Steve is going to call himself now, shows up with his team trying to figure out who Blonsky is working for and what the goal is. And the answers are "No one and nothing," Blonsky just wanted to draw out some Avengers to see if they could give him a challenge. Maybe the Hulk would show up and he'd get a rematch. Let Cap contend with someone who isn't working to some greater, awful goal. It would at least be a good starting point, if someone had perhaps suggested to Blonsky where he should try causing a disturbance.
Also, considering how Thunderbolt Ross berated Captain America for not keeping Hulk and Thor locked up, it'd be nice for the monster he stupidly created to show up and start wreaking havoc.
* To a certain extent, Frank Grillo's Brock Lumlow becoming Crossbones could have filled this role as well, although he did have an ideology he acted in service of. Irrelevant since they put him on screen for five minutes then blew him up in Captain America: Civil War.