I think it is a more complicated issue than it may appear. In addition to heal-from-0, the ease of casting healing word, death saves/dying rules, and the availability of raise dead (and freebie resurrection & re-builds in AL at low level), there's also the fact that in-combat healing comes from spells which compete with other spells for both known/prepped space and for all-important resource of slots, and that healing under 5e's fast-combat tuning can't readily keep ahead of damage, and that actually dropping a PC is about the only way to get across a sense of danger, since merely inflicting damage in no way impedes a creature in D&D.
Encouraging more pro-active healing by heaping punishment on the dropped character may not help much, the healer still faces the same resource calculations: slots are valuable, healing is not adequate to keep an ally up, HD, overnight healing, and heal-from-0 are all more efficient ways of restoring hps than proactive healing in combat, and he's not the one paying the higher price for dropping. :shrug:
If you want more pro-active healing, either introduce in-combat healing that doesn't consume slots, or make in-combat healing more potent, able to keep up with damage (which'll undermine 'fast combat'). Or both.
You can also reduce the 'efficiency' aspect by healing from negatives instead of from 0.