The first stage that the FM signal goes through is a pre-amp with auto gain control, next stage will be a mixer to reduce the received signal from its 100Mhz spectrum to an intermediate frequency. These will be in the metal screened can associated with the antenna connection.
So if the injected FM signal is many times higher than that of the received signal, the PLL section will lower the gain of the preamp and consequentaly the received FM stations.
Whats more, the FM transmitters are horrific quality so they will be far from broadcast standard and will emit a wide spectrum of frequencies which will mask radio stations close in the frequency range due to bleedover.
Stereo is also questionable as the FM setup is actualy for a mono channel (Left plus right), the stereo is produced by transmitting a further audio channel with L-R using AM supressed carrier at 38Khz above the base frequenc plus a pilot tone at 19Khz.
There is also pre-emphasis on the audio signal (they boost the amplitude of the high frequency audio before transmission and then reduce it in the reciever to improve the signal to noise ratio).
A simple low cost solution by the designers would have been to shut off the internal transmitter after a period of no audio input.
Similarly, the cost of a decent RF stage would run into the few pence to reduce losses in the device (I suspect that actualy there are very few losses and that the main issues is the AGC)