If you want to use a Photos Library on Macs with different system versions, you could use iCloud to keep the Photos Library on your older Mac in sync with the Library on your newer Mac.
If you want, try this:
- Get yourself an iCloud+ Storage plan sufficient to hold all your photos, if you do not have it yet.
- Then enable iCloud Photos on your Mac with your Photos Library and sit for the photos to upload to iCloud.
- Create a new, empty Photos Library on the older Mac and make this library the System photolibrary.
- Now enable iCloud Photos on your older Mac (with the option "Optimize Mac Storage", if you do not have enough free storage on that Mac).
- Your old Mac will now download the photos and videos, albums, and folders from iCloud and you will have a version of your Photos Library on the old Mac, that is compatible with the older system version. And iCloud will keep the two libraries in sync. Anything you do in Photos on either Mac will be synced to the other Mac.
But is "2011 iMac" and "Big Sur" really correct or a typo? I could not even my upgrade my 2011 Macs to Mojave.
If you are using some smart hack to run Big Sur on an unsupported Mac, iCloud may not work properly. Then you can still view your iCloud Photos Library in a web browser at www.icloud.com