SMS sender isn't generally something you can trust. If you get the SMS directly from the carrier that's responsible for the number, and you have reason to trust their SMS sending to verify the sender, then yes. But in countries with number portability, you still need to pay to lookup the carrier responsible for a number.

And you'll need to maintain ingress numbers in all the countries you support, and maybe numbers per carrier, depending, and you'll need to tell the user the right number to text to ... it's a lot, and it might not work well or might not save much money.