You're pretty much there. The key is that meat != meat, even in our world - a stingray stabbing your ankle and one stabbing your heart are both hitting "meat" but one is a lot more likely to kill you, and in the HP abstraction the one that got to your heart either got you while you were already worn down (low avoidance/luck) from something else, or you got supremely unlucky (critical hit, which bypassed a lot of your "luck" entirely.) Or even both. Either way, 5e would represent the latter scenario by the blow doing enough damage to take you to zero.
To me, the far easier way to address this (if it even needs addressing) is by implementing one of the grittier rest variants, rather than by trying to overhaul/bifurcate HP entirely.