We are in 2020, and at least it is possible to send a SMS using an Amstrad CPC thanks to Kiragregis. It uses an RS232 interface (by Amstrad) connected to a GSM A6 module bought on AliExpress.

If you understand french, you should really listen to the vidéo by Kiragregis, it's a funny one. If you dont understand, maybe the youtube subtitle will be good enough to understand it.