If the approach of "relying solely on the device to send direct textual messages" were adopted, the mainstream digital mode (D*Both /YSF/DMR and APRS can send SMS messages. As for the less common ones, I suspect they also support it, because sending text via digital modes is technically simpler than sending voice. However, D*It's quite unusual; he doesn't use "text messaging" in the normal sense, but instead adds text information within the voice data (of course, you can choose not to speak at this point). YSF is hidden deeper, in both the Log options of the menu and the GM mode. As for DMR, there are too many models, so it depends on how the manufacturer implements it.
As mentioned by the older gentleman, DTMF messages can be converted into APRS message format using an APRS repeater/gateway with the APRStt function enabled. For receiving, the repeater/gateway converts the text to speech. (Otherwise, it would be very difficult for humans to decode DTMF.)😇)
In a broad sense, as long as there's a modulator/demodulator (either built-in, modified, or external), any data can be transmitted, and text is no exception.