Having listened to the ATC it highlights the clear problem in the USA of controllers handling multiple frequencies without band-boxing. It's almost impossible to make a radio call without stepping on someone because you can't tell if someone else is transmitting. My policy in the USA is that if I need to make an emergency turn I'll do it and wait for ATC to contact me. Not being able to get a call in clearly added significantly to the crew's workload in this instance which isn't ideal in a terrain heavy airport with the performance issues associated with being single-engine on departure at high elevations.