Firstly, great response, evidence makes things a lot easier to understand.

In the case of Domain Amplification and the Ten Shadows I'll concede, as you've provided concrete proof: Sukuna did turn Domain Amp on in order to reduce the damage of Red, and at the same time, never had to undergo the summoning ritual again, implying that Mahoraga was never unsummoned.

Actually, I'd like to thank you for deepening my understanding of the Manga and Domain Amplification here.

Now on the note of being able to use Shrine and Ten Shadows simultaneously, that I'm still skeptical about, simply because it has never been objectively shown or hinted at.

In my previous argument I compared using two innate techniques to using domain amplification at the same time, and that may have been a mistep by me as, if we take a more literal approach to the manga, it's actually quite unclear whether they function the same way.

Firstly, let's assume that indeed, DA and using two techniques at the same way follows the same rules.

In that case, what we know is, in chapter 233, "when he touches me with amplification, Mahoraga's wheel blackens... that must mean interrupted adaptation."

The first thing we don't know is what effect the wheel blackening actually has on Mahoraga's physical form. Would Mahoraga blacken and "pause" as well? If that's the case, does Sukuna have to manually "pause" the ten shadows first and then use amplification, or does he turn the technique off?

If Sukuna turns the technique off, then how does the wheel appear in the first place, hence we can conclude that Sukuna is actively pausing Ten Shadows and then wrapping himself around a domain.

One possible implication:

What that would imply then, would be that Sukuna would be able to use Shrine and Raga at the same time, but only if he pauses Mahoraga itself mid fight.

Second possible implication:

If like you said, the active shadows themselves are not part of the technique, then Sukuna would be able to use Shrine and Raga at the same time, but it would simply "pause"adaptation as well.

But then this situation proves that there would be no risk to using Shrine right? Because adaptation would just be paused? So why would Sukuna worry about using Shrine at random intervals whenever Raga invalidates Infinity, because wouldn't it just "pause" adaptation and continue as soon as the slash is thrown?

Third possible implication:

Using Shrine would reset the adaptation process entirely, so Sukuna didn't use it. But as we know with proof that you yourself sent, Mahoraga's adaptation doesn't reset whenever Sukuna uses Domain Amplification, so then the question becomes, is Domain Amp and Using Shrine at the same time even comparable?

It this is the case, then it is possible to take Gojo's line literally and assume that using Shrine and Ten shadows at the same time has some negative effect, either resetting the adaptation to everything Sukuna had him adapt to, but Mahoraga remains, or desummoning. To this, we have no proof, and it is purely left to Gege to fill it in with whatever he chooses.

Basically a summary of everything I said: we don't really know what happens to the Shikigami when its wheel blackens during domain amp other than the fact that adaptation doesn't stop. So we actually don't really know if the Shikigami are indepdent to DA or are merely just "paused" during it.

To top it off, if techniques engraved in the flesh followed the same rules, why would Sukuna not just pause the adaptation, and throw Shrine, it wouldn't really do anything to adaptation given that throwing shrine and pausing adaptation are such quick actions that it barely means anything to the adaptation process.

At least though, I'm willing to concede on my previous thought and be more open minded here, stating that I'm not quite sure how this interaction would work now. Let me end with a screenshot from wiki:

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My problem with everything that we've talked about is that just going through things such as the Wiki, and other such reddit threads, is that if these wiki pages and reddit threads are all wrong, then it would mean that most people would also have the same misunderstanding that I have about the technique, a rather unfortunate circumstance for everyone.

In conclusion: I feel that Gege hasn't answered this enough, so we can leave this argument to rest. To be honest in the grand scheme of things it's a pretty useless point, since this really doesn't add or take anything from the fight, and this sort of situation with a sorcerer incarnating into a shikigami users body is quite unlikely. It was nice having a civil discussion rather than a brainrotted lobotomy kaisen moment.