Well I don't think its about mastering your doujutsu. Itachi, despite attaining the MS, could not possibly be the only to have mastered his normal Sharingan. Yet we never see his father Fugaku or anyone else having it on all the time.

Also, there is no Hyuuga that also has their Byakugan switched on 24-7. I think its just a trait of having a large amount of chakra or a large source of chakra. Remember, even though Itachi had his on 24-7, when he fought and used his Sharingan extensively he had to switch it off (e.g. the time against Jiraiya).