Maybe I'm way off, but I think it's helpful:
- internally the Atari directly generates composite video, and audio, two signals that contain one image;
- the TV modulator takes the video and audio and combines them into a radio frequency ('RF') signal, one signal that could contain all the terrestrial channels, which is supposed to look to a TV indistinguishable from the input it'd get from a real PAL transmission received via a real aerial.
An unmodified Atari has only the modulator output — that's the thing that comes out of the closed metal box that probably has a red sticker on top. You'd plug that into the aerial socket on your analogue TV and tune to the location in the pretend radio spectrum that the Atari was pretending to broadcast on.
You could reasonably expect that to do something at least with any TV built prior to 2012. In 2012 the final analogue transmissions were switched off so a newer TV probably doesn't have decoding circuitry for analogue PAL. It'll still have the aerial socket because it's still expecting to receive radio transmissions, it just thinks they'll be Freeview digital.
If your TV has an RCA socket, it's probably expecting composite video. Unless your TV has very peculiar wiring, it's very unlikely that plugging an aerial output into an RCA socket will do anything.
A VCR should have worked since they all need to be able to demodulate RF in order to be able to record it, and usually offer SCART output, which carries RGB, composite, and a bunch of other options. So aerial to VCR; SCART from VCR to TV is a normal arrangement.
I cannot think of a reason why a TV might be able to pick up audio but then lose it, for all time. But if you're looking for a hardware modification, I'd dare imagine that just extracting the audio and sending that to external speakers while muting your TV would be the easiest thing.