The page implementation in google tag manager just sends the page properties into dataLayer without adding an event key or anything that would make this usable as a GTM trigger.

I think it could possible add an event key like page_view or maybe just not implement page at all as I don't think there's a standard format for sending page views - GTM just generates its own page view on load and doesn't provide any way to send fake page views that it would treat as a "Page View".