SMS submission is now available in ODK Collect v1.16 and documented at https://docs.opendatakit.org/collect-sms-submissions! Thanks to @joeldean and all who have participated in the conversation.
There are a number of additions that are being considered for this including auto-send, functionality to let the enumerator choose the destination on submission, and functionality to make it easier for enumerators to re-submit a form sent by SMS to a server once they have connectivity. If any of these features are important to you, please open up new topics in the features category to discuss your use case and how you would use them.