Well, Novice should always be available, otherwise you would be pretty screwed at the lockpicking tutorial. Not to mention being required to magi-learn how to use the skill before you even can use it would be a bit of an immersion break. Especially when you go to trainers and buy lessons in Lockpicking, only to discover you can't use anything you have learned because you can't even pick a novice lock. Makes no sense.
I would say that those specific perks should become two-tiered starting with Apprentice locks. Novice stays as it is, but further on, and you will need to first pick up the Apprentice, Adept, etc. perks if you wish to even be able to pick that lock (at a slightly reduced skill requirement, though) and the second tier would be what the perks currently do.
Gives you reason to invest in the skill, as well as some extra benefits. This would, of course, need to be tied into a reduction in lockpick quantities, because as said, they are too plentiful right now.
I pretty much agree with the two tier system but with an additional restriction (lock jam).
The lock jam is basically a limit of how many attempts you can have with a lock before it jams and can't be picked again unless you have the same tier perk i.e. you've jammed an Adept level lock using up all of you 6 tries. You need to unlock the Adept level perk you unjam that lock. Locks in populated areas are unjammed after 24 hours by NPCs.
Get rid of Novice perk.
You can pick Novice and Apprentice without any perks.
You need Apprentice perk to try for Adept. It also greatly reduces Novice and Apprentice lock challenge, these locks never jam and you can pick Apprentice jammed locks.
You need need Adept to try for Expert which follows the same principles.
You need Expert for Master.
Master does the same and also automatically unlocks up to expert level locks.
The number of tries per tier before jamming is (with same tier perk/without perk):
Novice (12/8)
Apprentice (10/6)
Adept (10/6)
Expert (8/4)
Master (8/4)