Because said champion is balanced with the restrictions they have. If Lux did not use mana she would be impossible to push out of lane and would be nerfed. Riven has been balanced around her energy/manaless kit. She has been nerfed because of it. Just as Lux has been buffed due to her mana-using kit.
If you want to compare a champion that is in any way like Riven but uses mana to Riven herself, I am all ears. But you cannot come to me asking me why a champion like Lux, who is ranged harass and does AoE damage from a huge distance - uses mana and Riven, who has to jump into a middle of a teamfight to get her damage off - doesn't. There is no way Lux would be balanced without restrictions, just as if Riven saw a resource added to her kit she would see buffs.
And she does NOT need those.
As for your 2nd point, if Riven wastes all of her dashes running towards you then she's going to have her E, W and AA's to do damage with. Just punish her for wasting her main source of damage to get to you? Definitely not a moot point. The main point.
Yes, Riven mid is a thing. Elaborate? I don't understand why you added that. S:
how it could possibly be fair for 1 champion to have less restrictions than the rest of them.
I answered that with all the reasonable posts you could dream of having on the boards. Still on topic. :] If you aren't going to understand that this is a poor example and that Riven and Lux's cooldowns both make sense to their specific champion kits, that's fine. But with all the evidence I've supplied to make sure you see the facts of the matter I don't want to sit and just argue, so if that's what you want then I suggest you go somewhere else.
I'm all up for a conversation, though. :)