They should be sleep deprived the same way for it to be a real control group, at least in the context of "becoming a father". Otherwise it's just "being sleep deprived for 6-12 months has X effect", which is much less informative. We already know being sleep deprived for long stretches is really bad.

but then you're not comparing what it is like to be father with what it like to not be a father.

such an experimental design would miss the forest for the trees.