I’m pretty sure, at some point in European history, a style like that existed. But, I’m equally certain it existed as a contingency, to keep a combatant alive long enough to replace their weapon if it was lost or destroyed in combat.
Fighting using only a shield is in the range of comic book nonsense and Dark Souls challenge builds. Not something you intentionally want to take into combat.
The shield exists to provide protection from weapon strikes. I mean, that’s the entire philosophy behind it. While you could just grab a shield and bash an unarmed opponent in the face, it’s not the entire reason to carry one.
If your character is just obligated to bash people in the face, then a truncheon or baton will allow them to do that, while being easier to carry, lighter, not obstructing their vision, and remain usable in tighter quarters.
This is just a “right tool for the right job” kind of situation. Just because you can use a shield to pound nails doesn’t mean you should.
If your character is facing armed opponents, then a shield alone isn’t going to be enough. Again, the idea is you have a tool to deal with your opponent’s weapon, but the shield itself isn’t particularly good at filling both roles. At that point they’ll need another weapon, whatever that may be.