I keep hearing that the the lockpicking skill does not change how difficult (or easy) it is to pick a lock. Is this really true? If so, what is the point of having the skill increase at all?

It does make the locks easier to pick.

However the whole problem is, anyone can pick any lock at any skill level, if they know what to do. That is what's wrong with Lockpicking. I should only be able to stare at a Master or Expert level lock at an Apprentice Level Lockpick skill. Remember, it's your character that is picking the lock, not you. If your character is ham fisted, then they are not good lockpickers. Anyway, once you know what to do, the skill level ups are meaningless.

Lockpicking was done right in Morrowind. It has not been done right since.

The perks do make it easier but they are pretty useless as anyone with controller skill can open a master lock with 15 lock picking skill.

So it is useless but only to those who can play the mini-game. If you have problems with the mini-game, use the perks.