Why the prevelance of mechanical oscillators in electronic circuits?
Because the mechanical devices are much more stable than their electric counterparts. Let's compare a crystal oscillator to an LC oscillator:

Crystal:


Has a very high Q. According to wikipedia, a crystal oscillator has a typical Q of 10,000-1,000,000.
Stable with temperature. Many crystals are specified at &lt;50ppm over their temperature range, and temperature compensated or controlled crystals are also available, down to ~1ppm with temperature
Manufactured to a tight tolerance. Cheap crystals are usually specified to ~25ppm, but tighter tolerances are available


LC or RC:


Not available as an integrated device, so must be assembled from off the shelf components (unless integrated into a mcu or similar)
Low Q, it's difficult to make an inductor with a Q higher than a few hundred
Temperature sensitive - making temperature stable inductors is difficult
Voltage sensitive - the threshold voltage and charging voltage in the feedback circuit is usually voltage dependent. 

However, that doesn't mean that electric oscillators are never used, just that they're not used where great precision is needed. They do however have some advantages over crystal oscillators:
They can be easily integrated into another IC. Many microcontrollers now come with an integrated oscillator
They (sometimes) use less power. Often times a microcontroller will include a low power oscillator to run the watchdog timer, which uses less power than a high speed (MHz) crystal, and sometimes less power than a low speed (32.768kHz) crystal.
Since they can be integrated onto an IC, they can be used in places where a crystal would be far too large
They can be tuned fairly easily. A crystal can only really be shifted a few kHz off its calibrated frequency, but by adjusting the capacitance of the LC circuit (like with a varactor diode), the frequency can be adjusted over a fairly wide range. This means that LC oscillators can be used in circuits like PLLs or VCOs, possibly even locked to a crystal reference.


Non-mechanical oscillators are used in many devices, just not in those where accurate timing is required.