Triadic, I am not suggesting your switch one radio off (or have the volume on 0%). I consider it commonsense to monitor both, but use only one as the "active". I routinely hear taxiing calls on Area after I have "changed to CTAF", and I have even on occasion got a traffic alert from ATC about a recalcitrant aircraft doing something silly. Active volume 80%, inactive 50%. I can even tell what frequency they are talking on. And if I was SP IFR, I'd do exactly the same.
Having, as an SOP, that it will be normal to run two radios (presumably, you have one and the FO has the other?) is, in my view, asking for trouble. I would be not at all surprised if that, or something similar, is a factor in this accident; certainly the running of both active freqs so close to the airport increases workload dramatically. And that's what Class E will bring with it.
As far as briefing goes, there should be no need; it's not that hard. If a crew has to specifically brief how to run the radios in a CTAF, they shouldn't be there.