I think you may be wrong about illegal transmitters stopping car keys from working. I believe that the keys work on a frequency of 433.92MHz and use a super-regenerative receiver. The general use of super-regens went out shortly after the invention of the superhet in 1918 because they would pick up anything in the waveband. Radio amateurs are allowed to transmit 400W between 432 and 440MHz, although 10W is the norm, but even this would flatten the mW that a key would produce. The band for amateurs is classified as secondary, therefore there are other users also on the band.

I think that North American cars use a lower frequency but have not investigated what that clashes with. Cars have had poor radio immunity and I believe that the AA were kept busy towing cars off the M1 close to the Rugby radio station (now closed) and the Sinclair C5 would jam its brakes on with a small amount of radio energy. There are illegal transmitters and since there is now a charge for investigation, these have less reason to worry.