I have text conversations with my therapist regularly. Sometimes just about appointment times but also to clarify issues or as a means of support ( but I don't see T every week).
However from the other side of the fence I am also a nurse. I have frequent text and/or email communication with my clients both for appointments and support too depending on their personal choice. There is nothing to say it is unethical as long as it remains professional - for example I would not send a joke as I might to a friend. Neither text nor email are secure, (but then neither is a telephone call) but both can be later accessed if required by a court- even if deleted off the computer or phone. I print out emails and store them in patient records. Text conversations I transcribe into records.
So I would say it is not about whether texting is appropriate - it is about the content of the messages. If you have asked your wife and she has given a reasonable explanation then I see no reason for concern.
And I don't care what the American College says, firstly we are not all in America, but also technology moves at a much faster pace than policy. Everybody texts these days. It is normal life. Here in rural Scotland text is more reliable than voice calls as we have such poor coverage by mobile signal providers.A text gets through when a voice call doesn't. Most of my clients prefer texting, and it means they can reply at their leisure.
Oh - and although I don't see him regularly my psychiatrist ( yes - doctor not my T) does EMDR and Brainspotting. He does therapy - not just meds.