Link or select content from another site

I would like to define content on a central site (which has no front-end and is solely for content), and allow editors to select and use this content on other sites running on the same instance.

When I click the "Link an existing page" dialog and the "Link Page" sections open, there is a "Site" dropdown. However, this dropdown only contains the current site that I am on.

How can I make this dropdown show other sites, please?

Upon examining the CMS code, I noticed that it sets a "Where Clause", which filters to the current site name only.

It seems that the dropdown is only populated with other sites if the dialog is in "Copy" or "Move" mode.

Is it possible to achieve what I am attempting? I could modify the code in the CMS Project to enable this functionality, but this approach is not ideal, as any future hotfixes could override this change.

Alternatively, I would like to be able to switch sites in either the Page or Path selector dialog in the Page Builder.