They were referred to as DTA’s - https://en.wikipedia.org/wiki/Digital_television_adapter

You’ll still require a means of amplifying and broadcasting the converted signal if you need true OTA but this gets you an analog RF signal (NTSC in the US) from a DTV broadcast.

I wouldn’t be surprised if you could take the output of the DTA and feed it to signal booster/amplifier with an antenna on the output and get short range OTA broadcast. Just know your laws on allowed output power.

That coax output you mention contains the OTA signal, but yes, you’d likely need to amplify it within the limits of your local laws to broadcast it any usable distance.

Plenty of those exist. They were given out like mad when the analog to digital switch was happening. Little boxes that would convert digital TV signals to analog for viewing on old TV’s.