EDIT: Turns out there is probably an issue with the page. This exact problem is described here with possible causes: https://threewill.com/working-with-and-around-sharepoint-2013s-minimal-download-strategy/
I believe start.aspx is required for MDS to be able to pick up a diff between two pages, however the clients' browsers are being redirected to the page without it, so they aren't able to see the benefits.
E.g. client hits this page:
After 2 seconds, they're redirected to this page:
Trying to get rid of that redirect so that MDS can do its magic