Lux has a certain set of problems and then after buying her item, functions how she is supposed to. It fixes her issue, and she then works, her ability use or it's power is not dependant on mana. Mana was something that you could said came up as related in her given problem, but it's not the cause. It's not something that restrains her performance.
In other words, she's broken in a place but then fixes herself. (See also: "the feeling that they are disinclined to want to put that much effort into balancing ")
Champions whose performance are restrained by mana and would be broken by this hypothetical item in question are the ones like Kassadin, in his various versions.