Reminders priority feature

"When I try, it doesn’t seem to have any effect on how my reminders are organized."

That's correct, it doesn't, at least not in iOS. The priority markers (!, !!, !!!) seem to be there for purely visual effect, as far as I can see, just to attract your attention. And these markers have no effect on sorting your reminders in a list. In fact, there's no sorting option in iOS Reminders at all! But in the Reminders app on macOS desktops and portables you can sort by priority. Hope this helps you! Regards, Ian.

You're most welcome!