Kishi brings in the Byakugan in Part 2 at the least expected moments like at the kage meeting.
We know a lot about the Sharingan’s drawbacks(blindness, visible strain in the eyes like blood etc..) and some things about the Rinnegan’s drawbacks(time delays with strong gravity jutsu, using the revive jutsu while on low chakra could kill you etc..) but when it comes to the Byakugan’s drawbacks we know NOTHING.
The Byakugan has interesting nick names I remember from part 1 “Evil Eye and All Seeing Eye” which is a reference to whiteness of the eye like white ice cold(evil). You may be seen a lot other supernatural stuff that associate white with evil it’s quite common. Neji earlier on in the series at the Chūnin preliminaries exam when he fought Hinata he was actually quite epically scary he was about kill her but was stopped by Kakashi, Gai, Asuma and Kurenai.
Also in part 1 Neji’s dad and Neji were watching Hinata and her father sparr and he couldn’t believe how noob Hinata was compared to Neji, as Neji is a branch family member like his father their role is to protect the main family which are labelled as the stronger clan members. As a result Neji’s dad had a evil stare of intend to kill towards Hinata. Hinata’s father senses this and used the curse seal placed on branch family members to paralyse him.
Note: Hinata’s father could sense the intend to kill via the fine chakra changes the Byakugan can detect inthe chkara network and associate this to emotions. By the way nothing in the series apart from fox mode can sense at this level.
If you can’t remember these events well I suggest you guys look at these they are actually good insights towards to the Byakugan and the Hyūga clan.