Bear in mind that "illegal broadcast" is important here. A pirate radio station does not have a license to play the music it's playing. If recording pirate broadcast is legal, then effectively all copyright laws are moot so long as the copyrighted material is transmitted over radio.

Tivo is legal because cable providers are legally transmitting their cable programming.

Things get different on Netflix because recording Netflix requires circumvention of DRM, which is its own can of worms.