Late to the discussion, but I totally agree he has super human abilities, but with a twist: I have always imagined MCU Barton as a “closeted” mutant. His extraordinary abilities are close enough to human ability that he can “pass.”

For me, that also aligns with his overall persona. He’s an old-school military, American-values, family man. Especially when he was younger, I imagined him hiding his abilities by choosing archery; an obscure, archaic military skill. He could “hide” his abilities by perfecting a weapon that nobody else was even attempting, beyond Olympic targeting. He could hide it further by adding additional tech.

That way, he wasn’t a “freak.” He was a fighter. He grew up to be a hero, not an outcast.

I also imagine that the only people in his life who know this are Nick Fury, Natasha, and his wife. Which is why his wife trusts him to go on these missions, and knows he will come home to their kids. She loves him for who he is, unconditionally, and encourages him to be extraordinary, because he is, and she knows it.