The Gentle Fist does not come pre-packaged with the Byakugan.
Yes it does, all Gentle Fist techs are an application the Hyuga's ability to use all 361 Tenketsu, an ability stemming from the Byakugan, just as the Amaterasu Arrow is an application of the Blaze Style, an ability stemming from the Mangekyo Sharingan.
Kaguya, the person with the strongest pair of Byakugan on the planet barely even used it offensively due to her lack of the Gentle Fist.
Vacuum Attack is a Gentle Fist technique, its basically a more powerful Air Palm.
The reality is, the Gentle Fist is an invention, an add on of sorts to the Byakugan by the user.
I know it is, it's still a Bloodline limit. Assuming Hashirama was the first person since Kaguya to get Wood Style, he too would have had to invent the Wood Release techniques he'd use on his own, that doesn't make them any less of Bloodline Limit abilities. Same goes for Kimimaro. He was the sole Bone user in his clan, so naturally this means all the bone-based techniques were invented by him. Same for Byakugan. Just because the Gentle Fist was invented that doesn't make it any less of a Byakugan ability.
The Sharingan's abilities are all pre-packaged. I get that you like the Byakugan, but you have to learn that the Gentle Fist and the Byakugan are two separate entities but work together as one.
Again, no, the Gentle Fist uses the Byakugan's ability to utilize all 361 Tenketsu.
Even if you're born with the Byakugan, you can't use the Gentle Fist automatically.
Even if you're born with Wood Release, you can't use the Wooden Dragon automatically.
Like, no duh, pal!! Of course the Gentle Fist needs training! Just because something is a Kekka Genkai that doesn't mean its available from birth!
It has never been implied that the two abilities are tied together besides the fact that the user can blend them together.
Air Palm, Vacuum Attack, Rotation, and One Blow Body are all classified as Kekkai Genkai.
Anyone can use the Gentle Fist
No, pal. Plain and simple.