The oscillator is a Colpits oscillator. The output is taken from the collector not the emitter, but that has little effect on the oscillation.
There are two issues. One is that the base-emitter capacitor is missing. Normally there is a capacitor around 50 - 300 pF between the base and emitter. For the oscillator to work, it could have been relying on stray capacitance within the transistor.
The second issue is that the 2N2222 has a transition frequency (fT) of 300 MHz. That means that it will have a signal gain of only just above 10 at 27 MHz.
The missing capacitor would be the one that provides power from the output of the oscillator back into the crystal to sustain oscillation.
I suspect that the oscillator design is heavily dependent on the secondary characteristics of the transistor.
If you can't find the exact type of transistor used, I would suggest an RF transistor, with a much higher fT, and maybe add a 50 - 100 pF capacitor between base and emitter. Of course, if you are trying to find if the circuit is oscillating, you need an oscilloscope that can measure 27 MHz. Just testing things on the basis of overall function isn't much good in cases like this.