In a lot of games, experience is used as a way to prevent new players access to some content.

Traditionally, this has also been used to keep experienced players away from starter areas, because it's just too boring to be there. Guild Wars (2?) had somewhat alternative approach to this, since everyone is leveled down so even for a high level character it makes sense to explore starter areas.

With the skill-leveling system that Elder Scrolls games have, where you level things the more you use them, why is there also experience and leveling?

Seriously, it makes no sense to me. For roleplaying purposes, I want to be able to use the same character in a starter area and have it make sense, as I'm using in a late-game area. And having levels and experience only causes complications with game balance and PvP etc. If someone wants to go into PvP or end-game content with only one skill proficiency, I say let them. And if someone want to just go back to a starter area and farm some boars, why make it as boring as possible?

tESO is still in beta, it's not too late, please reconsider this. Pretty please?

How exactly does it not make sense

How should a good system work according to you Sigurd Viking ?

^ This. Its a gateway to allow you access to more base stats. Its a solid and simple system used by almost every game since someone came up with the idea. The leveling skills is specifically for the skills and they have no bearing on your characters overall level. Sorry its not like the older TES games, but that system is far too fast and easy to work well in an MMO unless you turn it into a horrifying grind.

I'm more worried about the quests to get that experience. So very sick of, "Kill X number of Y." quests. I don't mind having to fill an experience bar if it's at least interesting. I can understand such quests in a starting area, but why does a "hero" need to be bothered with killing a few roaming boars or whatever? RIFT was a great example of this: "Hey an Ascended (that games version of hero)! Do me a favor and go pick those potatoes for me." I of course spouted some profanity at the nimrod that approved such a quest and rode away. I seriously hope ZOS has put more thought and effort into their quests and how experience is earned.