Thank you for this! That is exactly what I did - added layers to a map, then went back and made the layers editable after saving the map. I was very frustrated that Field Maps Designer will tell me that the map is read-only but not let me instruct it to be editable (at least not as far as I could find).
If anyone else ends up here from Google, I tried saving the map again, which didn't work initially. Next I tried to save-as to a new map, which was also read-only. Then I went back to the original, tried flipping offline access on and off a bunch of times, and that didn't work either. Eventually I disabled editing for a few individual layers in the map and then re-saved, and suddenly it was editable. I'm not sure if changing layer editability was necessary or if making any change to the map would make the re-save take, but if re-save alone doesn't work for you, making other changes to the map before re-saving may help.