This IMO isn't the problem at all. The problem is threefold:
- Sword and Board combat basically caps when your Str hits 20
- Str sword & board is no better than Dex sword & board at hitting people thanks to the rapier being as good as the longsword. And it's worse >90% of the rest of the time.
- Synergies and anti-synergies apply
First increasing your strength bonus by +1 has
roughly the combat effectiveness of either Great Weapon Master or Polearm Master assuming you have no other relevant bonuses - and no other feat comes terribly close (not even Sentinel). I can't be bothered to go through the math here - but what this means in practice is that there are three ASI equivalents possible for either greatweapon or polearm fighters and only two for sword & board fighters. This means that the sword & board fighter lags behind badly at level 8 when you get your third ASI unless you switch to one handed spear & shield/polearm master (and see below for why that doesn't always work).
Second the Versatile ability of the Longsword is a ribbon ability that almost no one uses (especially with duelist style). What this means in practice is that if you assume equivalent stats a Dex based sword & board fighter with a rapier will do exactly the same damage as a Str based fighter with a longsword. But (even assuming heavy armour) Dex applies to initiative, one of your best saves, and multiple skills including stealth. Dex >> Str. Further if you want Dex-melee there's a good case for going ranger not fighter for a whole lot of interesting choices.
Finally there are synergies - with sword & board being the basic "nothing to see here".
- Great Weapon Master really gains from easy attack modifiers, so it's awesome for reckless barbarians and for battlemasters
- Polearm Master requires bonus actions and benefits from static damage boosts so it's best with Paladins. The Barbarian uses a bonus action to enter Rage and the Fighter for Second Wind. And subclasses like Echo Knight and Rune Knight use bonus actions for their class features a lot
- Polearms (and especially PAM) work better with duelist than with Great Weapon Fighter. It's one of the many things that leads to the spear-and-shield combination.
But I think the big thing is that the sword & board fighters stop scaling with ASIs at level 6 for fighters and 8 for anyone else - and the great weapons and polearms at 8/12. And they don't just stop but might as well be dex based.