1.) Tribal-Mercenary culture, not a warrior culture, they glorify winning and succeeding on missions, not honorable combat.
2.) The Hyuuga, all of the clans really, had their own cultures and traditions due to the fact that the clans existed and operated independently for centuries, or possibly millennia in some cases, before Konoha came along less than eighty or so years ago. What makes you assert that Hyuuga were a warrior culture at any point?
3.) Talent for many things has little to do with birth and most of the clans still hand down headship through blood, hell the whole concept of branch and main house falls apart if you expect people not to make judgements based on bloodline and/or birth order. The fact that they have this separation means they already reject your position on this. You're right, but they don't believe that.
4.) A warrior culture requires trainers, innovators, managers, and such who can stay home and keep things running so the army can fight. Even if main branch do go into combat they could do so under heavy guard and only for important shit or 'safe' assignments so that there's no easy way to grab an unmarked Hyuuga.
1)Warrior culture is not the same thing as honorable combat.
The Vikings were often raiders and brigands, they did not stand around to fight honorable battles, but were just as likely to climb in your city at night and gut you while you're sleeping. The Japanese samurai during the Sengoku Jidai era literally breathed treason and betrayal, and did their very best to avoid anything remotely resembling "honorable combat". The Apache were adept at ambush and counter-ambush, and certainly did not announce themselves.
Nothing says warrior cultures do not value results over form. Hyuga are ninja after all.
2) Because in canon, strength and skill were the primary requirements for the Hyuuga, and why Hinata had trouble in her younger days. We see this reiterated when Hiashi was watching Neji fight in the chuunin exams.
Similarly, in this quest, the literal test of adulthood, to speak and be heard in clan councils, is a series of fights, not an endurance race, or the ability to learn a book, or to draw a seal.
3) Because heredity is stable.
And we have yet to see any clan head that has managed to hold his post without being the best his clan has to offer.
It's especially notable that Neji is regarded to be a prodigy for being the first time a branch house member surpassed a main house peer.
4) Yes. Those people are called dependants.
Valuable and as important as they are, they do not control the House, because you need to have the respect of the people at the sharp end of the stick.
There is a reason why literally every clan head we've ever seen has been the deadliest, nastiest specimen that clan has been able to produce in his prime.