Well, I think only 2 class really can think of using 1h axes:
1-Shielders: Most of the time, maces or picks are better than their short cutting counterparts. If a shielder is getting str mostly, swords seem better. If more agi, picks seem better. If balanced, I'd say maces are better.
2-Swashbucklers: Well, I think swashbucklers are good to go with anything depending on their stat distribution. Still, the same dilemma. Maybe balanced builds can make use of axes with some good STR values, as you said it scales well. However, STR also gives more knockdown chances for maces so I'd simply pick them. By the way, I really like thrust oriented swords as a swashbuckler though. I think the class excels mostly with those weapons.
Sidenote: Don't forget that pierce damage glances less at low damage ratings, compared to blunt damage. That's why picks are I think better for AGI builds.
Axes don't give that good damage compared to the reach loss. Also don't forget that people who wear light armor are faster, so it is almost always better to fight with a sword against them. Against heavy or medium armored, picks or maces are better anyway. Axes are maybe only good when used to assist teammates. Removing someone shieldless is not important in 1v1, but it can be useful in 5v5. People invest skill points to use a shield, and you simply negate that with a few blows. Shields can block more than 1 directions at the same time, but a weapon can't. Same goes for an arrow.
I don't really think axes are good for cavalry. They are a bit too short I think.
I think axes are fine balancewise. The fact that they can be switched, make them good all arounders too. What I propose don't have that much with the game-balance actually. It is more about game-design. Some low armor penetration rating (a little damage reducement will be needed of course) would make them maybe best suited against medium armor. For light armor, swords would be better, because of good speed and reach. For heavy armor, picks or maces would be better, because of higher armor penetration.