Attributes were totally superfluous because they never should have raised on level. They should have raised on SKILL level. As I swing around my huge warhammer, my strength should raise as well. Well, once you take that step to have them level at the same time, doesn't that effectively make strength = weapon skill (or whatever skill you had that was governed by strength).
This was abstracted via the attribute raise bonus. The system was poorly designed either way. I say that in the OP. But the solution is to fix it, not remove it.
Then what's the difference between having your perks emulate as IF you HAD high strength. For instance, once you reached a certain point in Two handed weapons, you might get a blunt weapon perk that lets you strike an enemy hard enough to throw them back several feet. That is a REFLECTION of the strength that your character WOULD have as a result of the all the hard work raising that skill.
If you are strong enough to send someone flying with a hammer, you are strong enough to send someone flying with an axe. If you do twice the damage with a high two-handed skill level, you should do lots of damage with a 1 handed weapon. Because your actual strength grows. Perks will, at best, represent a portion of your strength that grows independent of the others. That's why attributes are useful: they keep track of one thing that, with good design, should influence a number of other things. That so many attributes in TES games only influenced one other thing is a sign of bad design.
Its the difference between raise skill -> raise strength -> get perk based on strength that knocks enemies back OR raise skill -> get perk. You're just cutting out the middle step, its just that strength is now IMPLIED by ones skill level. Having a 100 in two handed weapons heavily implies that I am a heaping ball of muscles, because it would follow logically, and would be no different as if I did ACTUALLY have 100 strength.
I guess it's pointless to point out that skill with a weapon does not necessarily imply strength. Weapons are usually dangerous enough regardless of your strength, as long as you can wield them effectively. A master swordsman will usually kill an unarmed UFC champion because it takes very little effort to stick a sword in someone.