Question / Suggestion: Page media (not inline images) in a central folder

I know you can make a folder for media so that images that are linked within a page are not stored within the page folder itself, but what i don´t know (or possibly does not exist, which this then would be a suggestion), is how to add page media (at the bottom of the page content editor, where you can click to upload a picture, which is then also displayed as post thumbnail in the blog listing) and store this uploaded image to a central folder instead of within the page folder.

If this already is possible, please point me to the right documentation (which i have not found so far).

Start by examining the admin page content editor's page-media upload flow and the existing handling for media folders. Determine whether uploaded page media can be stored centrally while remaining the blog thumbnail, and document the behavior that would count as complete.

Written by the indexing model from the issue text.