Theory time! The Minamotos are half supernaturals.
Minamotos are inclined to like supernaturals despite being tasked to eliminate them for generation after generation. Tiara got attached to a Mokke, Kou is empathetic to half of the supernaturals he met, and even Yorimitsu, who told us there was something wrong with his emotional attachment, took a liking to Hakubo: He stayed with the demon on his death bed, no humans in sight.
Usually that wouldn't mean much but oddities keep piling up.
The Minamotos have fangs, that's not something any other human character have. No matter how dramatic the manga can get, a living human does not get fangs.
Yorimitsu, the oldest Minamoto we have seen, had the bigger fangs of them all, comparable to Hakubo's and Yako: A demon and a fox spirit.
Minamotos are closer to the Far Shore than normal people after all. Not because 'they are born doomed' but because they have a mysterious sixth sense.
This sixth sense doesn't only makes them able to see supernaturals it gives them resistence to their influence. Even a supernatural as powerful as Mirai had trouble messing with Kou, who is not a trained exorcist.
We are told spiritual power is what give them this resistence but we see no other human have this: The Nagisa family, the Kannagis, they all have more spiritual energy than a normal person, and while that gives them the capacity to use spiritual weapons, we get no sign anyone besides other supernaturals have resistence to supernatural influence.
Speaking of supernatural weapons:
It is heavily implied this lightning cannot hurt humans.
We never see a person be hurt by a supernatural weapon, even someone as unexperienced to exorcism tools as Aoi and Nene were able to use lightnings without hurting themselves.
There is no control to Nene's attack here, she probably didn't even plan on using an attack, yet she wasn't bruised, there are no signs using that hurt it all.
Meanwhile Kou zapped himself. The 'demon vanquishing' energy that his staff creates can hurts him if it hits him.
So it seems Minamotos gain resistence cause they have both high spiritual energy and some supernatural energy: The two powers that give other supernaturals trouble combined.
Notice how Kako say 'supernatural' as if any supernatural will have resistence to his power but spiritual energy needs to be extremelly excessive to count?
Teru could open a gate between boundaries with a few drops of his blood, no tools needed.
The closest we have seen to this sort of power was with Yako, who is fox spirit born in a shrine, a mix of spiritual and supernatural power.
People may say Teru and Kou needing to eat a demon hand to reach the Far Shore trains means they aren't supernatural but even full supernaturals have trouble crossing boundaries during a severance.
What this moment implies is "Trying to become closer to supernaturals is something people have been trying to do for a long time"
And I have reason to believe the Minamotos had far deeper supernatural ties in the long days.
Yorimitsu was able to form a contract with a demon.
A contract have always been a supernaturals tool to take advantages of humans, we see it with kako, the mermaid, tsukasa, hanako... So a Minamoto being able to do that with without needing to offer anything in return (akane got the watch, nene had her mermaid curse halfed by hanako eating a scale) is wild.
It makes sense though, Minamotos are less supernaturals nowadays them they once were: their fangs are shorter and despite still having a lot of work to do, Hakubo calls them weak sauce compared to the Minamotos he knew while he was alive:
BONUS: There are some implications that Teru is affected by rumors.
This part is less airtight but hear me out.
Teru is the most beautiful boy in school, his introdution was not as his main role of exorcist, but as the prettiest boy in school, the well known school prince!
And his popularity is off the chart, he is treated as an idol! A prince! A pavillion was build for him, this pavillion is the most popular attraction of the festival, check out the line!
People bow down at his presence in valentine's day! He is the god of valentine's day!! Everybody loves him!!
Yet the second he is off school no one cares. We see zero admires, zero fans stopping him on his way to work, zero people commenting on how handsome he is. He is at most 'a pretty boy' the same level as natsuhiko when he is out, he is not an idol or a prince.
This becomes obvious when we compare him to Aoi.
Aoi had admires since she was a little kid, people have always noticed her beauty, to the point Akane had to build a habit of stalking her to make sure no one harrassed her.
Meanwhile, Teru was isolated. He couldn't spend time with people, but is not like people looked at him anyways. Teru is the only one interested in his classmates, watching them have fun when he reflects on his isolation.
But Aoi is always in the spotlight, even at her most isolated people will stares at 'the beauty queen', something they had done all her life. She can't catch a break.
There are no hints Teru used to have this attention as a child, and even as a teen, he loses it when he steps outside of school. His rumors of being the school prince is no longer applicable. Only Aoi's beauty keeps being treated as abnormal:
“what makes ppl think teru and akane homo for eachother!! they’re obviously so straight!”
son. homies don’t be sayin you’re beautiful when you’re just talkin 💔
be so real why’s that the first thing akane notices you CANT tell me that ain’t gay ✋😭
don’t get me wrong i’m a multi shipper all the way in tbhk/jshk but dude akane thinking someone besides Aoi is beautiful and it being teru is a another level of homo on its own✋😭 TELL ME IM WRONG💔