Why there is no a variable-frequency crystal oscillator?
The variable frequency crystals are supplied as oscillators.  They come in many flavours.

However, they can not function as you suggest.  To fine tune a crystal, one uses variable capacitance.

Quartz crystal frequencies and temperature properties depend on the set angle, and orientation with a precision of a &lt; 0.0001 degree with silver electrodes.

Here is an example of the process, starting with a Quartz wafer "blank" that has been cut from a crystal rod at some precise angle.   

Here is a variable frequency MEMS crystal Oscillator.  It is programmed like flash memory in a "Time Machine" before placement.



SiT8008BI-71-XXX-000.FP0000 is a field programmable device in the low power family. It   comes in the2.0 x 1.6 mm package, and can be programmed to support different combinations of the following:
Frequency: 1 MHz to 110 MHz with 6 decimal places of accuracy
Frequency stability: ±20 ppm, ±25 ppm, ±50 ppm
Temperature range: -20°C to 70°C, -40°C to 85°C
Supply voltages: 1.8V or 2.5V to 3.3V
Output drive strength: 8 different options for different rise/fall time

Cost &lt; $1 in volume