Is this a concious effort the player has to make to avoid using the other 13, or is there some mechanic that would actively hinder us from leveling the other 13?
I'm sure we can all conciously avoid using skills, but I believe a game is far more fun if the game itself is able to hinder your advancement without your compliance.
It's not so much a matter of artificially limiting your character, but simply playing the character you want to play and not having him do what you don't think he would reasonably do.
I'll play a variation of battlemage specializing in one handed weapon, destruction and conjuration. I know that my character will only be able to wear one set of armor, so he won't bother training up both armor skills. He won't learn two handed weapons. Won't be an alchemist. Probably won't be a pickpocket or blacksmith. I'm sure there are other things he won't do.
He will only level based on what he does do, so eventually I will reach a point where I'm not leveling anymore unless I branch off into those skills and activities I'm not interested in in order to push my level. But, since I don't want to spend my play time doing anything more than dabbling in alchemy or blacksmithing I won't do it. So my effective level cap and the number of perks I will get will be lower than the theoretical maximum level that assumes I do so much of everything that I max everything.
Really, your play style will limit your advancement simply because unless your objective is to max everything (in which case you can't complain about creating a god character) you won't push all of your skills hard enough to get to the theoretical cap. Do you really think that just in the course of playing the game you will use every single skill enough to max it out completely?
It's not as bad as people think.