It didn't break it in any way. You levelled up Major and Minor skills to level up. You could increase other skills but they did not contribute to your levelling.
How is that not broken? First, you're required to pick skills that will advance your character level before really knowing how the skills work and how you want to play. Then you can freely go and use skills that were not designated to advance character level, so you could reap their benefits without having the world respond to it. Go and level swords, armor, and smithing to 100, and have your character remain level 1... that's not broken?
It left no room for character development beyond the planned character design, as the skills you pick at the start are what you'll be stuck with. You can't role-play one type of character, then during your adventures have something unexpected happen that has some kind of life-altering epiphany, because you must stick with the skills you selected at the start. A mage that becomes a warrior after realizing his magic was no match for a dragon, a warrior that gets pulled into thievery after encountering the corruption in Riften, etc.
I don't think it was an good idea to remove attributes as they created more difference between races but the new leveling system is much better than the old.
Attributes, as they were in Morrowind and Oblivion, were rather flawed and I say they're better gone. For one, they created a feedback loop with your skills (level Skill X to raise Attribute Y, which buffs Skill X), and beyond that they mainly just increased Health, Magicka, and Stamina.
I'm sure an attribute system could be made that Works Well, but Morrowind and Oblivion's system wasn't it, and as said, it's better gone.
You could always make a custom class.
Doesn't change the fact that you were stuck with a preselected set of skills for the entire game.