Rule Lawyer Groupie
We had the same attitude, until we re-read the death saving throw rules more closely. You then realize any attack gives automatic death save failures. We lost a 9th-level paladin shockingly quickly because of AoEs and follow-up attacks.death/fear. 5e PCs seem too robust to my tastes. No more save or die, and no real fear of really bad creatures, like rot grubs, or level draining.
We feel a bit more vulnerable now.
Damage at 0 Hit Points. If you take any damage while you have 0 hit points, you suffer a death saving throw failure. If the damage is from a critical hit, you suffer two failures instead. If the damage equals or exceeds your hit point maximum, you suffer instant death.