SOLVED!
I have found the issue. I had a non default Cypher Suite order specified. Setting this back to default, resolved the problem.
It appears something changed between October and December which caused my custom ordering to become incompatible with the way Flight Simulator connects after authentication.
I wouldn’t recommend touching these settings unless you are having the exact problem I described and even then, I would probably suggest using a program such as IISCrypto to change them, unless you are familiar with editing the registry.
https://www.nartac.com/Products/IISCrypto/FAQ/what-registry-keys-does-iis-crypto-modify
Edit: Thanks to the Flightsimulator.zendesk team for sticking with me on this.