Would Apple Mini 'work'.... Define 'work'? sorry, a bit facetious, part serious.
I'm not a Mac person... so can't answer how well OBS would work on new M1 based Mac (if at all) vs older Intel based models.
However, I can definitively say a $500-600 new Windows PC would work today, (assuming low-end CPU and GPU requirements, ie not high end gaming, lots of fancy effects in OBS, etc). Would such a PC work in a few years at higher resolution.... depends. For some people, and what they are doing specifically, a $2->3K Windows PC is barely enough. For others, an older PC is fine... it depends. there are a lot of variables to real-time video encoding/streaming that can go from simple/low-end hardware requirements, to upper end.
As for the new Mac, I believe it uses an ARM based CPU (M1) vs Intel based which OBS is written for. See other forum posts asking for OBS compiled for M1 chip.
I can tell you that a new Windows PC with 6core/12thread CPU, 1650 Super GPU or better, with 16GB of RAM from a Tier 1 vendor would run around $700-800+ tax, etc and should work well without having to worry (and become tech knowledgeable) about optimizing OBS for reduced hardware demands. Can a lower-end PC work, sure, but now you trade off $$ (for hardware over-provisioning) for your time and expertise in compute optimization, OS & OBS settings, etc. your call
Again, settings not saving speaks to another issue
Though regardless, that age of a system is going to struggle with even basic real-time video encoding, regardless
Others will need to comment on a recommended Mac config that would work for you (and they will probably want more details on what you are expecting out of OBS t answer, as again... large range of possibilities)