Ian
If time permits yes.
The correct immediate response is to aviate however. The other guy is ensuring I use the correct procedures before he/she does anything else. Shortly, hopefully very shortly thereafter ATC will be advised. What if the frequency is saturated and it takes a number of seconds to change the squawk and have it recognized by ATC and hope they have an open frequency to move any conflicting traffic? Surely you don't expect me to counter the aerodynamic laws that are taking effect if I don't fix the problem. What of a single pilot ship such as a citation? Should that pilot put off the ingrained stall recovery procedure to talk on the radios to inform ATC before recovering from a stall? No, aviate, navigate then communicate. Immediate responses are trained for a reason.
Sometimes you have to act and understand that there's no guarantees in life. Hopefully the big sky, little airplane theory works in everyone's favor. In my situation, had I waited till ATC responded and moved airplanes if there was a confliction, I would have been headed downhill anyway, in a stall however.
Literally sometimes all you have a 2-3 seconds to react. Only once in my career so far have I had to do something like that. Given the few years it took off my life, hopefully the last as well.
You can argue it all you want, but that's what going to happen. If its any consolation, I agree with what you've said for almost all situations.