If you launched /Applications/Time Machine.app, that would give you a series of stacked Finder windows and sidebar full screen where you can go back in time, select the candidate file, and restore it. The Finder Quick Look and Spotlight work in this mode of Time Machine panel. It is not however, for launching applications, but selections for Restore.
It is somewhat puzzling that Time Machine ran a couple of days ago and yet only a three-year old version is found on Time Machine.
So, did you reboot your Mac and there was no change with this manuscript not opening? Is the most recent version on that Time Machine panel only from 3 years ago?
I cannot emphasize the combination of using Pages and Time Machine together as there have been more than one report of a Pages document becoming unusable literally overnight. I haven't personally experienced it, but once Pages decides it is not going to open a Pages document, that is the end.
Here's what I suggest. Download and install the current LibreOffice 7.6.2 Suite, and see if it will open that Pages manuscript document from its Open panel. LibreOffice is not a Pages editor, it will open it read-only, but from there, you should be able to save it as a Word .docx and hopefully recover your content. If successful, make a copy of that Word document, and then open the copy in Pages again to convert it to a Pages document. Manually save it. Force a Time Machine backup at that point.