You sir made my day! Much appreciated.

Glad it could help someone!

Absolutely, I should’ve highlighted this a bit more in my reply. Indeed if Xcode lets you build/archive without switching to manual code signing, then you can simply export the .ipa through Organizer! Thanks for reminding this!

As @scalo said in a later response, this applies if you cannot build/archive your app. If you can indeed build and archive, you do not need to switch to manual signing. Simply follow along starting from step 4 here :)

Same over here, Xcode preferences shows my teams as expected now! They must've fixed some kind of back-end issue.