I've got two Macs, one of them is a few versions newer than the other. They are both currently running the latest version, but the older Mac has lots of OS and system files that the newer one hasn't
I did a search using the following keywords: mac remove old version
The most promising result described three applications that could work for you.
I suppose that backing up to Time Machine, doing a clean install, and then restoring from Time Machine wouldn't work. The backup would include all the garbage files that you were seeking to be rid of.
Maybe it would be doable to do a clean install, but then only restore selected applications from the backup?