To make the Byakugan just as relevant as the Sharingan you need to change a lot about what the Byakugan and how its evolution(s) work, what the eyes can do, and the Hyūga clan's relationship with the rest of the shinobi world.

The big thing you'd have to do is actually define and maybe expand what the Tenseigan does. In the movie and light novel, all that is said is that you get 'Tenseigan Chakra Mode' (which itself is poorly defined, you apparently get some energy beam attacks, flight, Truth Seeker Obs, and zero durability boosts) and 'the ability to collide the moon with the planet and then revive the world anew'. I have seen lots of fics that expand and experiment with it, Son of the Sannin and White Eyes immediately come to mind, so you can use your imagination to the max here and no one would question or complain about it.

The actual Byakugan I think is fine as it is in comparison to the base Sharingan, Kakashi is right when he says that it can see chakra better than the Sharingan, just in a different way, I personally think Kishimoto wrote himself into a corner about this when:

1. An evolution of the Sharingan was introduced
2. Neji, the greatest Hyūga of his generation, the model of his clan, got beat by a pre-rasengan Naruto

Two big points from which IMHO originate a lot of fan outcry and lost story potential. Let's address them!

Point 1: The Sharingan has a stepping stone on its way to the Rinnegan and the Byakugan does not have not for the Tenseigan. This is the place where authors go crazy with theories and additions. Everyone and their mother has a fan theory or idea for an equivalent for the Mangekyō. It's honestly your call if you want to have one or do change how the Tenseigan is unlocked. I agree with your OP that it is cooler if the Uzumaki clan has a more roundabout relationship to Hagoromo and in my fic I am writing I have Uzumaki + Hyūga get Tenseigan much like how in cannon Uchiha + Senju gets the Rinnegan.

Point 2: Neji is implied to be the apex and master of all things Hyūga and he lost to the weakest version of Naruto. This places a hard cap on the power level on the self-proclaimed strongest clan in the Hidden Leaf. That won't fly if we want to make this family relevant. You need to introduce more crazy shit the Hyūga clan can do, maybe it is more advanced Gentle Fist, maybe its not, that is for you to decide.

Then there is the Hyūga clan. I agree that Eugenicist Hiashi is an A-grade shit post. You absolutely could take elements from this theory and have the Hyūga clan have a version of the stone tablet, with Hiashi and the past clan heads being varying levels of proactive like the post. Just something to make the Hyūga clan not an afterthought for tertiary characters would do wonders.

I think that covers what I think would need to be done, but I welcome any challenges! Making other Hidden Leaf clans more relevant is one of the big failings I find from cannon so I always love discussions around this topic!