You describe how they should work in regards to setting…but it appears that the failure point is that the date won’t advance on it’s own..if I’m reading this correctly.
Many of these calendar disks are not very robust. I’ve seen many that over time, lost the little “teeth” that drive them off of the cams, and / or they’ve just lost enough tensile strength ? to hook up correctly.
Sometimes the cams wear down, and can no longer engage the disk correctly.
With the dial off…and the calendar disk in place, play with the advance and setting mechanism and see if you can look into the works from the side with a loupe, to see if there’s anything really obvious.
The only other option I think..would be to attempt to source another movement that you know has a “sound” assembly.
Can you share the calibre ??
Good luck !
Randy