The first circuit has the transmitter at the output, its the transistor configuration with the inductor and capacitor on the collector connected to the antenna. The term's RX and TX mean receiver and transmitter respectively.
The frequency drift can be a problem. You should add a circuit for carrier detection, it can reset all outputs, and the toy (car) stopped, don't go away when carrier lost.
There is a potential problem in your original FM transmitter circuit. The tone output of the dialler IC is much too strong for input to the FM transmitter. You need to reduces it using resistors. Please follows the weblink posted by steveM for an example.
Also C3 across the transistor C-E pins is 10n which is 10,000pF. This does not seem right.