Vertically center text on a page-document wide

Nope, not document wide. You could do it manually on a page by page basis or if the content is suffciently small that you can make each page a fixed height row of a table you could use the Table middle setting for the paragraphs.

With a tool like Framescript, you could set up a procedure that would determine the bottom gap from the last line of text to the end of the text frame and then split difference to push down the top of line text, but this requires a bit of coding.