RC controller design for small car
The 433MHz band is a public band in some countries, where any form of radio interference can appear at any moment. Therefore to begin with, you can rule out any form of analog radio, since such solutions might pick up any interference and treat it as a control signal.

Using a digital radio protocol means you'll either have to use:


a MCU
or some encoder/decoder integrated circuit
or a radio chip with built-in protocol support (basically a built-in MCU)


The 2nd alternative is less common nowadays, manufacturers like to make all such ICs obsolete.

The simplest form of digital protocol involves the analog value and some checksum. It might be wise to also include the dip-switch value as an unique identity code, if you wish to prevent the remote from car 1 to control car 2 etc.

You have to ensure that your cars are not set to work on the same channel. Likely you won't have many options to configure radio channel on some cheap crap radio like the one linked, but at a minimum you need different frequencies for the different cars. 

Make sure it is not some wideband crap that smears itself all over the 433 band, because then this won't be possible. Look for modules that have a fixed channel, or the possibility to set the channel. Ideally, one channel in the 433 band should be 25kHz wide, to minimize the chance for interference.

In most countries you are only allowed to transmit up to 1mW E.R.P from 433 to 434 MHz so you need to check this too. Beyond 434,040 MhZ, you can use up to 10mW in most countries. In other countries, any use of the 433MHz band for RC cars would be illegal, as the band is reserved for shipping container RFID systems etc. Where are you located?



Regarding battery life, you could sample the joystick at given intervals. Make sure there is only supply to the joystick at the point when you are reading the value. A simple transistor or MOSFET could be used for this.
Similarly, you could power down the radio when it is not used. Keep in mind though that the crappy Ebay radio might need some time to lock on the correct frequency.