Well, rule of funny shall always prevail, but really you don't have to ignore the facts, just bend them slightly.
In the first movie, Steve wants to join the army to stop bullies (specifically Germans) and has no real reason to single out Hydra. It's not until he learns that Bucky is captured, possible killed, that he breaks away from his role as chorus girl and ride into battle. After he saves Bucky, that's when he truly joins the war and at that point it becomes all about targeting Hydra.
Then in Winter Soldier it's impossible to say how far Steve was willing to go when it comes to taking down the Hydra elements in SHIELD if you removed Bucky from the equation since he learns about the two in pretty quick succession. However, we do know that Fury's original plan was only to eliminate those in Hydra but to leave SHIELD standing while it's Steve who says they have to bring the whole thing down. During that scene it's also Fury who brings up that he hadn't known about Barnes, making it clear that at least part of Steve's motivation in the total destruction of SHIELD is because of what has happened to his best friend.
Bucky is straight up the catalyst behind Steve's actions in Civil War. Right from the beginning when Crossbones mentions him. No stretching needed, that just happens.
So, yeah, the joke is based on a exaggeration because there is humour in exaggeration, but it all does happen to some extent. It has to have, or it wouldn't be funny.
And yes I just explained the joke. I'm sorry.