Is it possible to make a coil-less simple Shortwave Receiver
Yes, a circuit such as you've shown can emit rf. It functions similar to a pierce crystal oscillator...something for you to Google.  However, without "old days coils " to form a tank circuit, there's a good chance it emits not only at 10.7 MHz but on 21.4 and 32.1 MHz as well.  I won't go into licensing aspects here.  
A receiver can be as simple as a diode, capacitor, and a good ground and antenna.  No coil.  Of course, it will receive only the strongest signal and no way to tune it.  On the other end of the no old days coil design, a fast A/D, a processor such as a DSP, sprinkled with software, you can have what's called a software defined radio or SDR.  


Coils have their place.  They're not something that goes out of style.