The lossy compression on phone calls might prevent fax from working. Also, I don't know if you can route call audio from anywhere other than the microphone.
HQ audio quality (over 4G) is markedly better than land lines, and the fax protocol has had error correction since digital formats were adopted in the 60s and 70s. https://en.m.wikipedia.org/wiki/Fax My Palm Pilot (with a CDPD backpack) and candybar 2G Nokia could both send FAXes. I suspect there's no demand.
Fax has an ip protocol called t.38 however it is inconsistently implemented. Most services still rely on g.711 - essentially uncompressed voice - for transmission of fax tones. However there is no guarantee the end to end call will be g.711 even if specified during call setup. There is an app called eFax - unfortunately the service is not free but I as a subscriber routinely fax over the phone when it’s the only way to communicate (like with a doc office).