I was not actually aware this was possible, but looking through the archives of the community I have encountered multiple posts by people who had various issues with this functionality but, implicitly, they were using Alteryx to send text messages. Examples here and here.
The very simple setup described seems to be one where you use a [phone number]+[SMS gateway domain] structure as the 'mail' recipient. This is much, much simpler than alternatives also described here in the community (e.g. this).
After some work I think I have identified the required SMS gateway domain linked to the phone service provider of interest, but my flow fails when I attempt to send messages in the manner outlined. The error is not consistent, I have got 3 different errors, all linked to the email tool: "Failed to send to one or more recipients", "Timeout was reached" (after 2 minutes the flow failed - the other error occurred after 1.3 seconds), and "Failure when receiving data from the peer".
Example screenshot:

(In the obscured part of the to-field is my 10-digit phone number (country code 45 + 8 digit Danish phone number).
I was not 100 % sure if the gateway domain should refer to '.no' or if perhaps it should be '.dk' for me as I'm located in Denmark. But it makes no difference, the flow fails either way. I'm assuming there may be missing infrastructure elements/accesses outside of the workflow causing problems, but the errors Alteryx provides do not really give me much to go on.
Any relevant knowledge, ideas or suggestions are welcome.