How do IR rangefinders accomplish centimeter precision without high speed / high cost parts?
There are two methods I am aware of, three if you replace light with RF.

1: A simple circuit with an IR emitter, transmitting a short columnated pulse at a slight angle to the centreline of the measurement device. This refects of the remote object and comes back slightly offset. Capture the reflected light, measure this distance from the centreline, then the distance is simply geometry.

2: Uses a local oscillator, generating a sawtooth wave at a few tens of MHz. We transmit a laser pulse at the bottom of the triangle and trigger a sample and hold to capture the voltage when you get a reflection which is detected with a transimpedance amplifier and a photodiode.

The voltage + pulse count is directly proportianal to the distance.

Light takes 50ns (approx) to travel 15m and 1000ns (approx) to travel 300m.
The detector needs a local oscillator of say 2/50ns = 40MHz
We need 15cm accuracy, so lets make this 1 bit.  We need to encode each 15m into one sawtooth and there are 100 x 15cm of these, we need a 7bit ADC that can sample at 40Mhz. 

The counter needs to be able to count to 300/15 = 20 at 40MHz to achieve the full distance.  The actual distance is limited by receiver front end sensitivity, output power (safety concerns) and the problems with carry chains on fast(ish) binary counters.

Each 15cm time interval is seperated in both time and voltage, so capturing it should not pose a problem.

The last component is a sample and hold.   The circuit would require calibration to remove the error caused by the sample and hold's trigger delay. Other than that none of the components are expensive.

It is just as possible to use a very fast counter instead of the sawtooth, (and these do exist) but would be much more expensive.

3: You can substitude an RF transmitter for the laser and a directional antenna + RF front end for the reciever, otherwise the circuit is the same. (radio and light travel at the same speed)

4: By using RF you get another method, called CWFM, where you use the sawtooth to FM modulate the transmitted signal.  The received signal is mixed with the transmitted one, the output of the mixer is a hetrodyne (frequency shifted representation) of the distance, an FM demodulator can turn this into a meaningful signal.