I find it fascinating that Tesla would even be able to implement SMS over wifi, as it seems less than straightforward for the carriers to implement it.The thought has been going around in my head for the past day or two, and I asked the lady at the Bristol SC when she rang this morning, but she didn't know. Maybe it's like my wife's phone, as that sends and receives SMS texts via WiFi (mine doesn't, same phone, different provider).
Whether a provider allows SMS over WiFi seems to be really hard information to pin down. I know my iPhone can do it, as it's identical to my wife's phone, so I spent an hour or so yesterday trying to find which contracts included SMS over WiFi and which don't. The providers seem remarkably reluctant to make this information readily available, for some reason. I may just switch to O2, as we know that some of their contracts provide it here. The only two providers worth using locally are O2 or Vodafone, anyway, as they seem to have the best local coverage by a long way. I used to be with EE until we moved here, but EE's coverage around here was really poor, hence the switch to Vodafone.
The use of SMS certainly explains why waking the car seems extremely sluggish at times (problems with the mobile provider delivering the SMS in a timely fashion), and a lot more sluggish in recent months (before the not-waking-at-all problem).