One-handed, block, heavy armor, destruction, conjuration, alteration, restoration, enchanting, and smithing on one character isn't really limiting yourself. I think it is kind of wrong to be expected to use every single type of skill in the game on a single character. Heck, that character is level 55 and his heavy armor, block and restoration aren't even close to maxed out yet. I can't imagine piling on more into that list of skills. Eventually it'd get to a point if you just kept getting wider that all characters would go the route of the spellsword and become too homogenous.
It is limiting yourself if you choose to stop there. It's fair that you can't be expected to everything on one character. But giving more options increases what that character can choose to do as well as give more options for other characters. As opposed to simply letting that one level 55 level up some more.
And a raised skill cap implies new weapons, armor, and magic.
No, it doesn't. It implies a new material. Not new types. A new material of sword is not a new weapon. Throwing daggers are, however. And you can't implement throwing daggers by raising skill caps.
Unless the new stuff was flat out better, making older stuff obsolete, it would be more of the same with maybe a few unique gimmick effects like artifacts. Raising the level cap allows for a natural progression upward.
Not buying it. New weapon types and armor classes is only more of the same in a broad sense. And certainly in a more reasonable new content than just continuing the old trend upward. Having a new material of sword for 20 more skill points is not a more novel activity than being able to start throwing daggers and handaxes.
We don't even need to progress upward. So I don't see that as a benefit. We need more things to do and more character options.