Have the same problem after getting a M1 MB Air with Big Sur. Photo stream works for 1 or pics then stops. iCloud photos is unchecked. Have to toggle the prefs on both Photos apps. Then it works for 1 or 2 more pics. I have a MBP 2017 that is also running BS. Photostream works most of the time. . Please keep us updated with your progress. thanks.
Update: After numerous tries (resetting wifi, network prefs, iCloud, etc), to get this thing working, I found that after taking one or multiple pics, then accessing Photos, the pics would not load on PS from my iPhone 13. Then, toggling ONLY the Photos app My Photo Stream pref, It loads the pics. This works on both, my 2017 MBP and the M1 MBA. Hopefully, they come up with a fix.