Most people can probably agree that the current way abilities and attributes work allow you to be a little too flexible with how many skills you can learn.
When your char lvls up, you have to chose between weapon/defence/school, whereas in fact you should only chose between offense/defense. Schools refer more to skills (as something you can do, not a game term), it is more like thievery, telekinesis, crafting. I think that skill system should be splitted other way, by physical and mental nature of the skill. One-handed, two-handed etc is more about physical and learning abilities or advancing in some particular shool is more mental.
I don't think I fully grasp what your idea for the new skill system is.
I think you're saying something like skills are specifically tied to one-handed or two-handed, but can't be used by both. Even if I have that part right I don't understand how that would work for magic.
I also don't think that you have offered a compelling reason to switch away from the current system of skill-granting schools to a different system.