I agree with you on the logic front, but I think the extra attack is pretty much a Holy Bovine.
One desperately in need of becoming Holy BBQ.

It's never had any basis in reality but "gamers made up something that sounded right based on total misconception of how two weapons work". It doesn't even have the excuse of a lot of early D&D mechanics that otherwise get called nonsensical, but were based on a clear intention to abstract something more complex. A second attack is something specific, and fits D&D about as well as a called shot does--not at all. This is especially true given what even the six second round represents, as Ranger Wicket explained.
A better bonus for two weapons is a modest bonus to hit. Perhaps you can trade that bonus to hit for a bonus to defense (melee attacks only)?
In fact, that could solve the whole problem of a shield not being treated as a weapon, too. Set it up so that if you go aggressive with Sword & Board (i.e. bash with the shield) you get a bonus to hit at the expense of losing your shield bonus to AC. Set the numbers so that two weapons is slightly better on offense but can be traded for defense while S&B is the other way.
Given that D&D has always made the distinction between "hits" and "effective hits," it also wouldn't hurt in Next to swap all or part of the bonus to hit as a bonus to damage, depending on what the rest of the system needs to work--though this is of course even more abstact.
Then obviously 2-handed weapons get a very modest bonus to hit, too, which equally might be efficiently swapped for extra damage.
That leaves single-weapon/empty hand and complete empty hand. From a weaponry persective, these are distinctly sub optimal, which is why swashbuckling and the like are going to go for a second weapon or buckler whenever they can. To make these styles meaningful, you need:
- Useful things for that spare hand to do in combat that don't directly involve attack or defense.
- Special techniques learned by the character to make up for the sub optimal nature of the weaponry.
If we must have more of that "second weapon gives a second attack" nonsense, put it in a pure gamist optional module where it belongs.
