You know, actually Kagurabachi 117 might explain why the Sazanami ability 'Storehouse' is considered heretical (per Kyora) and also why Akemura zeroed in killing Hakuri so hard.
Long post incoming,
Like, look at the above old-timey propaganda. We know the Soga at the time of the war (and for a long, long time leading up to it) had an enormous influence on the government and were responsible for keeping the underground sorcery clans in check (mostly, it seems, by dangling access to the Prophecy Ceremony in front of them)
Magic Bad, Sword Good
Which brings something I've been thinking about for a while, for being a setting where magic is very real and can be very dangerous, innate sorcery isn't really utilized much. And when it is, sorcery is often just a supplement to physical prowess.
Take our canon strongest sorcerers in the Kamunabi,
Shiba: Guy who can teleport (something that is explicitly possible with 'anyone can learn it' sorcery, though obviously not to Shiba's speed level)
Azami: Magic coin trick that was thought to be so useless, all his ancestors did was use it for medical scams. He primarily operates by beating the shit out of people
As for the rest of the Kamunabi (excluding Hiyuki on the assumption that Flame Bone operates like an an enchanted blade), Uruha and Natuski, both have powers that they mostly use to supplement their swordplay ('enhance one physical ability' and 'volt-tackle', respectively). Kudo and Takefu have powers mostly good for support (fleeing and containing fights). Kiri doesn't seem to use sorcery at all.
While the Anti-Cloud Gouger squad is an exception actually seems to rely on magic more than physical prowess, (and have actual active combat powers), they are (1) lower ranked than above, (2) directly under Azami who is a bit of maverick, with no issues breaking tradition, (3) part of the younger generation that started operating after the war (they needed Chihiro to explain how Cloud Gouger worked).
Compare the above to the 'underground' sorcerers they fight,
Guy who can make bombs, guy who can manipulate mud to enough to make copies or sink a car, guy who can telepathically search for people and send them to to the trauma zone <- all just from the first arc chasing after Char
Or, the Hishaku. Hat Guy and Yura are always showing off with their pine trees of Fuck You, and prior to gaining a blade Hiruhiko seemingly relied entirely on his sorcery to fight. Even their swordsmen (Hokuto and Kuragi) seem much more willing to use and incorporate sorcery (even if for Hokuto, it's just for avoiding fights he thinks are boring), and have a much more over-powered ability (Seriously compare, "manipulate full suits of armor" versus "slight electric current with coins").
It seems like ever since the Soga Clan (Sword Guys) won that conflict and banished the Mikaboshi (Sorcery Guys), they and the government have maintained a culture of,
Swords and physical fighting = good guys, allowed to work for the government
Relying on your sorcery = bad guy, either stop practicing or work in the underworld
A New Player Has Entered the Area
So, if we assume that for a long time the Soga Clans maintained the status quo of Sword Guys in Charge (mostly use spirit energy for doping and maybe a little sorcery to supplement your blade work), and the more "powerful" innate sorcery abilities that popped up were quickly suppressed or could only be used in the underworld -
. . . what would make this leading clan, really, really pissed. Maybe pissed enough that they'd use their claim of a "divine" lineage to label something as 'heretical'?
What about some guy showing up (200 yrs prior to story start, so ~800 years after the Mikaboshi were exiled) with two wildly OP-sorcery abilities, and then passing both of them onto his descendants to create a new "powerful clan of sorcerers".
Isou is powerful in the regular sense, it creates a huge shockwave. It's a great innate sorcery to use in a fight. Notably the Tou and all the rest of the Sazanami family (save Kyora) use it as their primary weapon in battle. They strictly follow the 'bad guys rely on magic theme from above', even if Kyora does sometimes give them blades to yeet with Isou.
Storehouse, though, now Storehouse is wildly OP and impossible to counteract without sorcery.
"Registering" something gives the user complete control over it - you can teleport someone into a lava pit, into the endless void of 'Storehouse', remove them from battle, hide them away forever, etc. Only people who can use sorcery enough to notice it can resist it.
It is also (and I feel this would be really annoying), the ultimate fuck-you to weapons. Hakuri shows that while he can't register an Enchanted Blade without consent (as they are special and can basically hold a spiritual charge), regular blades - even those wielded by sorcerers are fair game.
You could understand why a clan of swordsman, might hate a guy like this? Right?
The Soga Clan whose self-professed job was 'controlling' sorcery clans, probably hated the Sazanami Clan's guts for being a 'powerful clan of sorcerers' that utilized both a strong attack sorcery and the ultimate no-sell to non-magic swordsman.
Unfortunately, -
Unfortunately, either the first Patriarch (or someone soon after him) was way more interested in Profit (The Raikuzaichi) than in sticking it to the anti-magic, government sword cops. And equally unfortunately, money will get you far in life.
It seems that prior to war, everyone was in a tense, "as long as we don't have to see it relationship".
Officially, the remaining or new sorcery clans "refrained from harming ordinary citizens and kept a low profile", but in reality it seems they simple had to operate in secret if they wanted to be invited to the table.
However, beneath the surface there seems to have been more tension than even that implies. Before the War, when Shiba's confronting the underground sorcerer who smuggled a piece of Datensaki, he equates Soga=Government, and that a sorcerer working for them is essentially their dog, which doesn't exactly scream good working relationship.
It seems the Soga/Government's agreement with the sorcery clans is only harmonious on the surface.
After the war, the Kamuabi seem to be struggling to got back to that same status quo. (Didn't include it, because this already has too many screenshots, but the example used in recollection of "minimal semblance of public order" is the Rakuzaichi). It's clear that the Soga (who are absent from the story besides Akemura in present day), were a big part of maintaining that old status quo, and that they maintained it by their power and legacy.
Conclusion (or TLDR)
The Soga Clan and the government were extremely close and operated to keep the practice of sorcery suppressed (seemingly since they exiled the Mikaboshi 1000 yrs ago).
Obviously they'd consider the Sazanami's stupidly sorcery OP-Storehouse ability heretical, and obviously Akemura's first instinct after wiping out one 'powerful clan of "evil" sorcerers' would be to go after the next 'powerful clan of "evil" sorcerers'
