An anecdote: my (not-IT) company does exactly this for out-of-band management... except, in one small satellite location, the phone company no longer provided any copper POTS lines; all they could do was an RJ-11 jack out of the ONT that was backhauled as (lossy) VoIP. So the modem couldn't be made to work.
My point being, it seems that modems are becoming less-and-less viable for out-of-band management.
Fun story, AT&T forced our hands to get off our PRI (voice T1) and move to their fiber service. They also insisted that they have a dedicated phone line installed so they can dial into their modem in case of circuit failure. We can’t buy a cooper phone line from them, so it gets routed over the same fiber circuit and goes to a digital to analog device back to the router. I don’t think one hand talks to the other over there...
My point being, it seems that modems are becoming less-and-less viable for out-of-band management.