I have found a fix.
Go to Settings --> Components in Xcode, uninstall the Codex component fully by deleting it. Then, after I waited for about 10 seconds, go in to the Intelligence menu, click Get on Codex, then sign in normally.
That resolved the issue -- must've been a stale install error with Xcode updates which persisted across versions due to the Codex component remaining installed.
Topic:
Developer Tools & Services
SubTopic:
Xcode