Yes, the lore has its flaws. However, some of this criticism can easily be debunked.
We know that the device has control over the minds of these heroines, and can bring them to Lex against their will and without them knowing.
Surely it's able to prevent them from leaking anything or exposing themselves to the truth in the daytime.
With that being said, it is also very possible that they are not sleep-deprived. While we don't know much of it, it could potentially induce REM sleep in the heroines, allowing them to get sufficient sleep.
I can't help but agree with destroyerofassholes in regards to disappearance during sleep and whatnot. This can simply be countered by making clones of the original heroines and replacing the actual heroines with clones. Tweak the clones to be significantly worse versions of the heroines and Lex'll have control of all superheroes, while the clones are failing the world. At that point he would have reached his ultimate goal with little effort.
Bruh, there's a far bigger flaw in Lex's system before all this harming stuff.
How the fuck does nobody notice that these girls are gone at times they should be sleeping, ALL of them together? Times when they should be fighting crime or if there is an emergency or if they have spouses that sleep with them how does nobody notice?
The answer is it's a game and we shouldn't think of its plot too seriously lül.
I know it's a game, but I'm probably among many who loves defined and well thought-out lores.