The Emergency Broadcast System was turned off in 1997, so of course cell phones don't have to support it.
They do support its replacement, the Emergency Alert System. That sends messages out as cellular alerts as well as the myriad of broadcast options, so your cell phone would get them anyway, even if you didn't have the FM app running.
Cell phones also support the CMAS, which distributes the same sorts of warnings as the EAS, plus a couple more. (Amber Alerts, for instance.)