When I create a new document with Pages v12.0 on macOS 11.6.5 and then Save… I see the following in the File Chooser side bar, despite the fact that Pages is deselected in iCloud Drive > Options… panel.
By saving the macOS Pages document to that iCloud > Pages location, it is immediately available to Pages v12 for iOS.
If you also see this on a Pages for Mac Save, then choose that iCloud Pages folder and then your iOS/iPadOS devices should also see the document. When I first launch Pages v12.0 (and this has been true with earlier versions), the only dialog that opens is the Template Chooser — provided I leave Pages deselected on that iCloud Drive > Options… panel.