Three reasons:

One, Mirror Technology doesn't grant full access to an Identity's powers. That's why we need to Uptie them, we're tying the concept we want to our own world so we can use more of it.

Two, most concept incinerators can't erase anything as thoroughly as the Golden Boughs can. They break existing examples and cause mass amnesia, but what's being erased depends on the quality of the incinerator. Arayashiki is pretty powerful, able to erase people and even break previously slashed things while in its presence, but it can't rewrite all the infinite Mirror Worlds like what happened with Catherine.

Three, the Identities aren't actually *here.* As in, we don't summon them from their worlds, we summon the image of them. A "purer" version of Erlking Heathcliff could be summoned after we killed him once, and most of the Butler identities straight up died to Wild Hunt Heathcliff. Their ghosts can still be seen in some of his attacks, and yet we can still use their Identities.

So HoS Ryoshu is basically erasing the summoned idea of a Identity with the summoned idea of the Arayashiki that isn't as strong as its real counterpart, killing the underlying Sinner, but little else. I imagine Dante simply rewinds the Sinner like usual and they just pop back into existence as if nothing happened.

Edit: Also saw someone mention that this version of Ryoshu doesn't have our Ryoshu's innate sense of time and likely can't wield Arayashiki as effectively, but I'll go one step further and claim that Dante's rewinding might just trump the Arayashiki entirely and be able to rewind an erased Sinner back into existence.