Yes, that is how it is supposed to work.
Utilizing the proxy workflow properly involves setting up projects using original 4k media, so Premiere will always ask where it is. When you initially open projects on your 2nd proxy-only system, then point Premiere to the proxies, then hit command-S, it should update where it looks for the proxies when you open up the project the next day (i.e. it will stop looking for proxies on your original system). It will still ask for the full resolution media each time you open the project, but that should be an easy one-click “offline all” move, and the project should open on your 2nd system with your clips online and linked to proxy media on that system.
@cyrusdowlatshahi