You're 100% correct, this is a backend issue with iCloud and not the device itself. Confirmed by a L2 support staff that is now escalated to Apple engineering Dept. First, there's two different yet related issues on this thread, I'll address the first only.
AirDrop "Contacts Only" bug/iCloud password prompt iCloud mail when composing is a backend service issue with your account with iCloud. There's no troubleshooting that can alleviate this problem from the user's end from the device in question. It will only to help diagnose the issue for now. So restoring, signing out and back into iCloud, and rebooting etc. will most likely not work.
Example, You'll notice a dummy iCloud account on the device in question that "Contacts Only" in AirDrop will work again as it normally would, but your real account won't. (Remember to delete your iCloud data on your device before using a dummy account so that non of iCloud data is uploaded to the test account.)
The device, in this case your iPhone or iPad attempts to ping the iCloud service with a password prompt entry on the device to enable "Contacts only" within AirDrop, it hangs a bit and then does nothing. You'll also notice that the bluetooth icon in the status bar is now exhibiting a new habit of disappearing and reappearing, when normally it would remain present on the top. "Everyone" works as white highlighted text, "Contacts Only" doesn't highlight when it should.
This bug sprang up when when I was experiencing an entirely different issue with contacts, calendars, reminders that were not syncing without manually pulling to refresh. Signing out of iCloud alleviated that sync issue to bring up a new bug the "Contacts Only" and "iCloud mail Password Prompt" issue.
Additionally as someone mentioned previously, you might notice that if you toggle off a service like calendars or contacts from iCloud on iOS settings in any of those apps and turning them back on will render the app useless or inactive and icons within the app appear grayed-out. Two ways to alleviate that specific bug is to either sign out of iCloud on the device completely and sign back in or add a "Test" entry within the specific app when it's toggled off and toggle it back on so that it prompts a "merge" to push back all your data back onto the device.
These software bugs and or quirks are more prevalent when the device itself is heavily tied cloud services, i.e. iCloud. So whenever there's a backend/server issue, expect your device to act wonky especially if the device itself unresolved software bugs. The growing pains of the cloud.
To help escalate this issue, ask for L2 support at Apple, describe the issue as best you can, and ask for an iCloud ticket to put your iCloud account in troubleshooting mode.
These are basic quality assurance (QA) issues to be focused on.