Using classic Outlook for Windows in business environments
Hi AnotherBug - Let me see if I can help clarify your question. You mention that Both Outlook 2013 and 2016 hare setup with the OAuth regkey. Those versions do not support OAuth. Here is the exact snip from the updated issue page. Per the below, you will need to upgrade to a more recent version of Outlook, or use the Outlook.com web experience in your browser.
For the Mobile device question, if you are running the latest version of Outlook for Android or iOS you are good to go, again assuming you set it up via the default config and did not do a custom setup syncing via imap/pop. If you are having problems with auth prompts on your phone, I would recommend removing the account and then adding it back using the autoconfiguration that it will do.