Anon, where you’re going wrong is that it didn’t stop with those three Tales of Suspense issues, the Clint/Natasha plotline continued on in a major way in the pages of Avengers after Clint joined the team. In fact, Natasha’s fate was a large part of what motivated him to join up in the first place.
Fate always exacts her price for past deeds! There was no way to escape— my darling! But, I’ll make up for what we’ve done! I’ll devote my life to making amends!
As you might expect Natasha doesn’t disappear forever, and the storylines she gets in Silver Age Avengers book are the ones that really solidify her character. And Clint’s a big part of that. I summed up the arc of their romance here. I wrote it about a year ago so I’m not totally happy with it, but I think that’s basically what you’re after.
I do think Marvel is putting way more weight on that past right now than they have in the past three decades or so, but I don’t think that means they’re sticking in stuff that wasn’t there. From an in-universe perspective, their relationship was a transformational chapter in the lives of both characters. From an out of universe perspective, their doomed romance was one of their main plotlines in their first few years of existence, when Marvel was still figuring out who they were and how they’d function in their grand corporate narrative.
But you’re right that their dynamic in the film is quite different from the comics! Largely because the filmic Hawkeye is not a carnival worker driven by jealousy to paint a giant purple letter H on his forehead. The idea of Clint and Natasha as partnered special agents, who’ve bonded over the mutual understanding of a lonely profession— that’s something taken from Ultimates. But 616 Clint and Natasha were two lost and screwed-up people that forced each other to redeem themselves.
(And I really hate what Deadly Origin did with the Tales of Suspense flashbacks— I don’t think they capture Natasha’s dynamic with Clint or with Tony very well at all.)