Well and good if it's all domestic flying and you have two areas to overfly...flat and mountainous.
On longer international routes, flying may pass a dozen times or more between high mountainous terrain and flat ground. We don't brief the terrain each time we may or may not be approaching mountains.
Come to think of it, I've never heard a briefing "when entering areas of high terrain."
To which regulators do you refer? No training program I've undertaken refers to 10,000 as an emergency descent altitude. None of the companies or agencies for whom I've flown used it, and the arbitrary altitude that we've always used at Flight Safety and Simuflite courses has been 14,000', excepting where higher terrain or MEA intervenes.
I grew up and spent much of my early flying life in places where terrain was well above 10,000. In fact, density altitude where I learned to fly was often above 8,000 to 10,000', and the idea of predicating an emergency descent to that altitude seems a bit odd to me. The actual descent altitude specified in my aircraft operating manual, directly from Boeing, is 14,000, or higher as required.