It’s all about the API. In FMP the magic happens with the Insert from URL script step.
What Connect gives you-and its no small thing-is ease of connectivity. You can replicate all of this yourself in FMP. If you have the time and interest you can create a robust solution for Twilio interactions. Couple it with a web viewer and you are set.
GitHub is a great place to start searching for source code. You may not find FMP code but you will find solutions in lots of other languages. That code could form the wireframe for your own work.