Can aviation HF radios receive while transmitting?
In general, the HF radios in aviation work in simplex mode.

According to the Annex 10 to the Convention on International Civil Aviation- Aeronautical Telecommunications Volume V Aeronautical Radio Frequency Spectrum Utilization Chapter 3, 


  3.1.1 In the aeronautical mobile service, single channel simplex shall be used in radiotelephone communications utilizing radio frequencies below 30 MHz in the bands allocated exclusively to the aeronautical mobile (R) service.


Though some aviation radios are capable of semi-duplex operation, virtually all aviation services operate on simplex.

I have no idea why the operator was stepping on the aircraft's transmission though.