We recently wrote an article comparing the infrastructure setup of various communication API platforms and how that affects their reliability. We'll have to see if Twilio says this was from their network or some other internal problem. https://voximplant.com/blog/where-cpaas-deploy-their-network...
For Twilio accounts, there is an option to add a TOTP secret, and use it when logging in. But, they don't support disabling the default SMS method, and using TOTP exclusively. When you authenticate using TOTP, you still have a fallback option to receive a SMS code.
So, in the case of Twilio, the TOTP feature only improves convenience, not security. At least that was the case when I looked a couple months back. I opened a support ticket about this, and they acknowledged the problem, and that was it.
