How to push changes to a channel to multiple layers using that channel as a mask?

[Ofnuts]

If you want to make that a mask for each layer, you'll have to reset the mask of each layer one by one.

But if you want to make all other layers transparent, you can set the selection to the mask, invert it, and then apply a script such as clear-layers to delete the selection in other (visible) layers. The selection being partial, the deletion will be too, so your bit with partial opacity should be kept.