Field is not read-only. Query works fine (list of choices is correct). However, upon Update the selection is not saved into the profile. Why?
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As I've had to do with other similar cases in the past where a field doesn't behave as it should, I deleted this one and started over defining it from scratch. I copied the query and re-created the field and it worked fine. This should probably be investigated.
That can happen if you change a fields type. This causes it to have to restructure its databased stored data to the new type. This doesn't always go smoothly. Running the "Check Community Builder User Fields Database with strict column types" tool in CB > Tools should catch these issues.