Crystals are a very high Q series resonant circuit, i.e. they posses an ESR of 10 ohms at their resonant frequency. If you shunt them by a small capacitor 50pF or so, they have a very high impedance, like a parallel tuned circuit, with a Q of 5,000, but you lower their frequency by several hundred hertz. So straight away you can see that the bandwidth of the circuit will be 27MHz/5000 ~ 5KHz, but to achieve this you must very loosely couple your aerial into the circuit, say with a 10pF capacitor. You have to make sure that both the RX and TX crystals are for the same frequency.
Strictly spoken you can't simply "replace the LC part" by a crystal:
- the transistor would miss the DC bias
- you don't have a means to couple the aerial with suitable impedance matching.
Furthermore, each LC circuit has a characteristic impedance and Q according to it's purpose. It's not clear, if the crystal will meet this requirements.