As someone who loves playing anivia recently.
No. Anivia has a lot of hidden power, and once learned she is one of the best mages currently in game, even if immobile. If you did this, She would need nerfing. There was a reason they didn't fully un-managate her, but the little they did makes her a strong laner.
Granted, Anivia's learning curve is extreme, and learning her takes falling on your face a ton of times, but once learned, she can stomp most lanes pretty hard, or evens out against her harder matches. For example, she has huge aa range, making for safe last hitting, even if the particle for it is meh at best. Her ult is the last on demand AS slow in the game, and her wall is the only on that can cover an entire lane. Her stun can double proc it's damage, meaning that her 2 main damage abilities get a 1.0 ratio for AP. Her poke is insane, assuming she lands her stun. Her passive is one of the best in the game, as it one of the most powerful baits in the game, all game long.
Heck, even if you get trolled and thrown bot lane as support, she dominates that lane. Walls and Stuns in bot lane are OP, not to mention AS slows.
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