I tried the last answer, with some mods because my situation is simpler - we have only two phones, one parent, one child.
(Sorry I can't find your handle, but it's the one where you used four fingers to tap on four phones simultaneously - you rock! The reboots - no help. I was very reluctant to reset all settings, and it turns out I didn't need to. So, thanks for your post!!)
This was working fine for many months, through at least a couple ios updates. Then a few days ago screen time was shut off. When it was turned back on (attempted to at least), I had this problem.
What worked - this is pretty fresh so I hope it sticks - is:
1) shut off screen time on both phones
2) shut off connectivity on parent's phone (wireless, bluetooth, cellular)
3) on child's phone, turn on screen time and down time. I modified the time to 10-5. Used my secret passcode there
4) on parent's phone, turn on child's screen time and down time. I modified the time to 11-5. Used that same secret passcode on this phone (does it matter? I dunno.) I also went into content and privacy restrictions, and turned off "allow" on Share my Location - this is a different thing I haven't done before, and I mention it because it may be relevant, the touching of some other setting.
5) waited a minute or so (again, relevant? not sure)
6) turned on parent's wireless, bluetooth, cellular.
And within a few seconds, the downtime refreshed to 10-5, which says to me that the child's phone's settings (icloud, whatever) rule the day. The "allow location" setting reverted back to on, also - the child's setting ruled. Trivia, doesn't truly matter, but it could help to debug stuff.
Interesting to note though that I was getting the "turn it on and it turns itself off again" behavior, with the Content & Privacy Restrictions button, when I went into the child's setting on my phone. I kept poking it, trying to turn it on, and eventually it started behaving as well, after about a minute.
Crossing my fingers that it stays working. This bug is a pain in the rear!