I did eventually fix it, but only by going through some steps with an Apple help person through chat. It was simple enough, just making sure things were toggled correctly and then restarting, but it seemed to work! I have transcribed the relevant info we did below, hope it helps someone else! Note: I have an ipad and an iPhone, so it mentions both. All relevant steps were performed on the iPhone:
Ensure all Focus features are turned off on your iPad then go Settings > Focus and turn on "Share Across Devices" on your iPhone.
Then from the same tab Settings > Focus > Do Not Disturb tap "Focus Status"
On that screen, toggle "Share Focus Status" to On
After that, go back to Settings > Privacy > Focus and make sure "Shared With" Messages is toggled On. Now toggling on and off the Do Not Disturb setting will work as intended and people should not see either the "notifications silenced" message nor "Notify Anyway" warning when the Focus is off.
If the previous setting is already toggled On, toggle it off and then back on.
Then restart, using this method, noting that there are no pauses or delays and all buttons should be pressed as quickly as possible:
Press and quickly release Volume Up button
Press and quickly release Volume Down button
Then press and hold the Side button for 30 seconds until you see the Apple logo
This resolved it for me. It is very specific, but I hope it can help someone else.