again, your connector does not convert anything as there is no "coaxial signal". A standard euro antenna cable is coax by definition just with a different connector. In Europe we used the round plug-in connector instead of the screw-on connector that you guys use in the US. While it is true that the screw-on connector produces a much more secure connection, there shouldn't be much of a difference in the signal quality. The reason why it worked in the case of the other forum member could be that he was using a busted cable? Just guessing.
The RF out was widely used in the 80s and early 90s on a bazillion computers and consoles. It was a pretty much standard way of connecting stuff to the TV. I don't see why it wouldn't work with the Atari.
My guess is that the RF modulator in the Atari is causing the problem or something else. I checked all the solder joints and they look brand sparkling new. Apart from a bit of dust, the PCB is also very clean.
EDIT: the voltage on the big 2200uF 16V capacitor is 10.18V and the voltage regulator outputs 4.91V which should be fine.