I am not questioning that without his shield Captain America is a great fighter without his shield. But something I often wonder is if a fighter like him is so used to having a certain weapon even it is just its mere presence it creates a certain comfort. That and how they used to fighting with it compared to fighting without it. Sort of like how Hawkeye is a good h2h fighter on his own, but is a far better fighter with his bow. Would you say that Captain America is even a tiny bit better with his shield or is he equal with and without his shield?
He's definitely at an advantage with his shield. He's mastered using the shield as an effective medium range weapon, a close combat offensive weapon, and it's an unparalleled defensive weapon.
Bare hands can only do so much.