[morzh]

Terry

Having tried to model it in my mind I do not see how not having Y-caps would hurt the reception much for transformer radios. In fact many 40-s and 50-s radios, both US and European lack the caps.

Let's say the chassis is not grounded. Any noise then induced upon the primary (yes you are right about inductance but then there is the through parasitic capacitance which common mode noise will use to get through) will appear as also common mode noise at the secondary. But since it IS common mode it does not matter much for the reception as it does not create any signal anywhere relative to the chassis.

Now if you do ground the chassis it is possible that what was common mode before after getting through the xfmr parasitic capacitance (and still is for everything other than the chassis which is now grounded) becomes a signal towards the chassis and thus gets amplified and heard. Partly this is taken care of by decoupling and bulk capacitance, though electrolytics are not really that good at radio frequencies being a bulk storage, so some low capacitance ceramics/paper/polyesters could help when used as decouplers.
Should you however use some Y-caps this becomes less of an issue in the first place.

Now in today's devices Y-caps are there mainly to protect the Mains from the noise generated by the equipment itself. I short it shorts out high frequencies between the device's common and the point where the Mains' earth enters it so not letting the noise escape in the Mains and cause interference for other people. This tale care only of really high frequency component.
Applying the same idea back to 30-s radios (though I doubt that was the intent, I think FRC/FCC at the time did not bother that much about these things) - well, the Y-caps can possibly filter out oscillator noise that would leak otherwise in the Mains.