Shields - Not For Blocking
"Also, you'd certainly be better off focusing on dodging an ax than trying to block it with a shield that's going to be hacked to bits."
This seems to be based on a misconception of what a shield is used for. Yes, in a shield wall, you use a shield for blocking.
But in a one-on-one fight, the shield is used to defend you in the same way a weapon is used - by controlling and guiding the opponent's weapon, not by simply "blocking" it. If you're holding your shield out and taking the full brunt of an axe blow with it, you're doing it wrong.
Dodging - A Good Way To Get Yourself Killed
"focusing on dodging"
Dodging is a very small part of combat, and is really more of a last-ditch defense or a way of sweetening a parry than something to be relied on. You always want to be seeking to control the other person's weapon, and you can't do that by dodging. You need to be making contact - either with your weapon or your shield, or by grabbing their hand/arm.
There's a rule of fighting - "Never be uncrossed in distance." That means don't *ever* let your opponent so close they could hit you with just a hand motion, unless you have control of their weapon.
Hollywood depictions of combat are terrible for this - the two fighters stand sword toe-to-toe and slug it out, constantly crossing and uncrossing their swords while they're close enough to punch each other. Well, in real life, both of them would end up dead doing that - being uncrossed in distance tends to lead to double kills.
Furthermore, "dodging" implies (to me, anyway) contorting your body in some way to get out of the way, something you should ideally not be doing, as it breaks your structure and makes you vulnerable. It's far better to use your weapon for defense than to rely on dodging.
But this is getting into the territory I'm going to cover tomorrow, which is Reach, and why more of it is better.
"That's a minor in-game justification, but since the goal is something minimalist that's in between the traditional Weapon vs. AC system and "all weapons are the same", it seems reasonable."
I respect the goal, but I think that any mechanic that encourages players to have their characters behave irrationally is broken.
You would most certainly be better off facing someone with an axe with a sword and shield than with just a sword. I think that really should be obvious.
I can see, if you want to give axes something special, maybe you could have them just negate the shield bonus (i.e. +1 vs. shield). There's no real-world justification for that, but it fits with the system you're building, and doesn't encourage bizarre character behaviour (i.e. "The orc has an axe? I quickly drop my shield and charge!).
A better approach might be something like, "unbalanced weapons count for twice their length", where unbalanced means any weapon with most of the weight at the end - mostly the mace and axe.
This would serve to model unbalanced weapons main function - dealing with armour.