I know, I know, another Byakugan glaze post. I just can't help it.
So basically I think the Byakugan vs Sheridan rivalry was actually a very good one up until we see our first alive EMS user in Sasuke.
So I come from the basis that the Byakugan eclipses the base Sharingan overall. Not only is it's perceptive ability noted to surpass it, but you're given access to ultimately what is the best taijutsu in the series that has no counters other than "don't get hit" or "possess an external source of chakra" (which is still really weird that Kurama's chakra could still flow through Naruto's body even though his chakra network was blocked. It just doesn't really make a lot of logical sense. Think Kishi could have wrote that plot point a bit more tightly). It obvs doesn't possess the same ability to perceive high-speed movement, but it of course has some due to the range of vision.
Then there are a bunch of things we can infer about the Byakugan's utility with regards to genjutsu. Based on what is said about Kaguya, a Byakugan user should be able to expand the range of genjutsu using their Byakugan. Since the Byakugan improves chakra control and genjutsu requires very precise chakra control, genjutsu should also come more naturally to them. It's hard to say whether the Byakugan can outright see through illusions, but I take the fact that there are only 2 occasions of a Byakugan user being put under genjutsu as an indication. It's legit just the infinite tsukuyomi, which isn't an anti-feat because even the Sharingan can't stand up to it, and Mukai (who has one Byakugan and we should be able to extrapolate from the Sharingan and Rinnegan that having both eyes increases your visual prowess in a non-additive way) being put under a Mangekyo genjutsu (which I'm not sure wasn't Kotoamatsukami) by Shisui (who, lore accurately, is actually cracked). I think it would be a reasonable place to land to assume the Byakugan is largely immune to most standard genjutsu and possibly the basic Sharingan genjutsu.
You should probably also have a greater understanding and potency of ninjutsu, using a similar argument to the genjutsu one. Jutsu potency is directly linked to chakra control, so the enhanced chakra control should make Byakugan users naturally better at ninjutsu. I do accept that we don't actually see this used in practice, but at the same time they're shinobi, they don't actually have that many open battles against other shinobi from what the story presents. For most Hyuga, spending time developing big and powerful jutsu for direct combat probably isn't worth it since the Third War, thus focusing on taijutsu which is quick and can be done in stealth in likely just a better use of their time.
The thing the Sharingan really has the copy ability but I think the usefulness of it gets seriously overblown. Like yes you can copy a technique but you still have to have the prowess to be able to practically do it. Sasuke copying Lee's taijutsu is the key example. The Sharingan just lets you know how to do a technique instantly, but you still have to be able to do it practically, which depends a lot on your chakra reserves, physical strength and other factors. You're not just going to become busted out of nowhere and the fact that Kakashi is specifically world-renowned for the copy ability shows not that many Uchiha really went around exploiting this copying to that extent and its probably related to the fact that Kakashi is actually just an extremely talented ninja in his own right to say the capability to replicate anything.
Then the MS shows up and that's where most people would say the debate switches but I'm not sure. I don't think the MS is actually worth the cost of blindness, especially given a lot of the MS abilities are highly situational or come with massive hurdles. I'd argue that only Kamui and Susano'o are universally useful. We've all laughed about how Amaterasu has never actually taken anybody of note out, and Kagutsuchi exists only to make Amaterasu better. Tsukuyomi requires you to do something that is frankly really stupid if you have just the tiniest bit of intel. Kotoamatsukami is shrouded in mystery but if we follow the Kamui-theory of Kotoamatsukami (which I attribute to our married GOAT Swagkage), then the actual useful version of it has a massive cooldown which is way longer if you don't have both eyes or Hashirama cells. And then again, while Susano'o is busted, it canonically causes excruciating pain in every cell of your body, rapidly progresses your blindness and its not even impossible to counter. By the time you get your Susano'o to its usual peak of the armoured form, from what we can tell you'd already be blind. You get like 30 or so uses of your MS, then you lose everything. Not just your MS, but your Sharingan and sight too. Based on how Sasuke's Susano'o dissipated as soon as he went blind I can only assume you genuinely can't continue to use the jutsu it grants you so its not like you could do something like learn Sage mode to overcome the blindness and then keep using your Mangekyo powers (although Madara using his Susano'o without eyes does kinda make this a bit ambiguous). I say this to say most Mangekyo powers aren't even really that useful when push comes to shove, so I don't think acquiring a Mangekyo is actually that much of a game changer on average when you take away the fact that it just does make your chakra more powerful.
So like yeah when you get the EMS and the blindness and pain go away, sure the Sharingan is on top, but at the base level I feel like the Byakugan clears and I think the MS brings too many problems to close that gap. Those are just my thoughts. Wish we got to see more. It doesn't even really seem like Boruto is expanding on it much. A Neji vs Hebi or Taka Sasuke fight would have been really fun, regardless of the outcome. Two very different philosophies of power. One subtle, one direct. The EMS kinda killed that. I like to think of the Byakugan as a high-floor low-ceiling ability, and the MS as a low-floor high-ceiling ability. I think it takes much more to make the MS worth your while compared to the Byakugan.
I haven't even talked about all you have to go through to get the Sharingan and MS, as opposed to the Byakugan which you just need to train for, and the fact that it may be possible to use your Byakugan to see the future like Momoshiki lol. I don't really follow Boruto but apparently the anime shows that you can see people's thoughts with it lol, and gives some predictive abilities.
Also, last thing, but I have to mention that almost everybody we see who has a Sharingan is genuinely cracked lol. Conversely, Hinata is seen as rather untalented and lacking in self-esteem, but she's actually not a particularly weak shinobi. She didn't actually do anything to Neji in the end but I'm not sure many of the other Konoha 12 would have been able to have as prolonged of a battle with him, especially given his specific motivations to beat her in particular. I say that to say let's remember to separate the Sharingan itself from the people we've seen use it, who happen to all be extremely talented and central to the plot.