I seem to remember reading in the 1970s or 1980s that coiling power cords is a bad idea because they turn into 50/60 Hz radiators.
No it turns into RF choke filter and reduce RF interference injected into mains with switched mode power supplies

Since power cord has two wires with current flowing in opposite direction, they cancel electric and magnetic field from each other, so emission will be almost zero. With reactive load there is some phase shift, so some amount of energy may be radiated, but not much, because phase shift with home electronics is too small. For high power loads like heaters it's almost zero.
So, in short, it's better to put cable into a coil, it will reduce high frequency emission which is present in mains from different parasite sources.
For better filtering you can use mains filters which uses coil with core for better common mode current cancellation, something like this:

If you're using radio receiver or transmitter, such filter is must have, because it reduces RF interference leakage through mains.