Teaching the 8 Gates to everyone is a very stupid idea. [I believe that] The 8 Gates is a technique that is only known to and fiercely guarded by Gai because he knows how powerful it is. His dad took on the 7 Swordsmen of the Mist using that technique. That is an incredible amount of power which he isn't stupid enough to just give to everyone. Torture exists and all it will take is one broken Jonin to lose valuable information and jutsus to other nations. Imagine a village like the Hidden Mist, you don't think if they had this technique, they would train a team of ninja to turn into a suicide squad to destroy their opponents.
Hell, I don't think we have enough charisma to actually convince Gai to teach us and the time spent on getting him to take us as an apprentice will be way too long.
There's a TON of jutsu like that in canon that literally no one is doing that with. Why? Who knows, but they ain't.
But more seriously, Gai blabs about how it works in public, infront of a lot of foreign ninja, and there are a lot of ninja around who have the skills needed (fancy eyes, or a high level sensor) to just look at Gai/Lee and figure out everything they need to know about the technique. It's a VERY simple technique, it's opening specific tenketsu. As I mentioned earlier, any Hyuuga could do it with ease. After that, all you need is conditioning, which literally every ninja works on (though maybe not to the extent of Gai/Lee).
[X] Plan: Itachi Peer
If we want to have a happier story with Itachi and Dog, we will need to be strong enough to have that happy ending (it is the Naruto world). Might as well use our genius to its best extent while still having a peer we can both support and be supported by.
Thing is, Itachi and Kakashi are also people who are REALLY skilled, and train A LOT. If we become friends with them, then we can train WITH them. Get Good Friends to Get Gud With Friends. Hell, Kakashi is going to become a Sensei in not too long, him teaching us can easily become his job if he likes us.
Plus, I'm afraid if we don't pick one of them this turn, we won't GET to be friends with them. Their curiosity about would have just dried up.
Uh...it makes perfect sense why the Hyuuga don't go into the Gates? The Gates basically bust the limiters open, which would make their entire fighting style obsolete, when they're all about precise, controlled Chakra application.
And also noting that even the first three Gates fuck your body up. They make you physically better in the short term, but at the cost of increased body damage, aggravated fatigue and more. You need to be incredibly healthy to begin with to avoid the Gates crippling you with the backlash.
This is opposed to the Hyuuga's entire strategy based around a superb sustain game where they out-defend their opponent, spend less chakra doing so than their foes spend to attack and keep stacking debuffs whenever they make a mistake.
Busting your limiters doesn't fuck over your chakra control. It shoves it all into your hands for you to personally deal with, rather than letting your autonomic systems do it. What's more, it boosts your speed by a
fuck ton. Which the Hyuuga ARE all about. That, in addition to precision, is the cornerstone of their martial art. Imagine if Lee knew the Gentle Fist in that fight against Gaara. Instead of Pinballing Gaara like he did, trying to break through his sand armor, he could just tap him a few times, and BAM, fight over.
Why do you say the first 3 hurt you? The second explicitly revitalizes you, by forcfully boosting your stamina.
The Hyuuga only play the attrition game when they can't just quickly down their opponent. The Gates would grant such a huge burst of speed and power allowing them to quickly bull-rush anyone who would have proven an obstacle otherwise, they only need to land a few hits to the right spots (sometimes even just one hit) to take out almost any opponents.