Apple ID password reset - an infinite loop?

Until you have access to your Apple ID again you can skip transfer and signing in to use for calls and texting.

Account recovery is a two phase process once started. Phase 1 is the evaluation period, usually 24-72 hours but can be longer or shorter. This phase says Apple is evaluating your info and you’ll get an update after the posted time is over (people mistakenly think it means they’ll be able to reset the password immediately after this first phase.)

Once phase 1 is done it enters phase 2 where https://iforgot.apple.com/ will give you the final amount of days (can be weeks) you’ll wait. On that final day you’ll get a text/call, follow that text/call immediately. If you don’t receive the text or call, once the period specified in the original email is over, you can still go directly to apple.com/recover. Follow the instructions to regain access to your Apple Account.

During the wait period make sure you are carefully following “After you start account recovery“ at How to use account recovery when you can’t reset your Apple Account password - Apple Support 

• “If you started your account recovery in Settings, System Settings, or the Apple Support app, you can continue to use that specific device during the account recovery period.
• Regardless of how you started the account recovery request, you should turn off all other devices that are currently signed in with your Apple Account until account recovery is complete. If your Apple Account is in use during your request, your account recovery will be cancelled automatically.
• If you started your account recovery request with iforgot.apple.com through your device’s browser, you should avoid using that device during this period. If possible, turn that device off. Using that device might cancel account recovery. "