Honestly, it's a real problem with the 5e dying rules in general - to make things scary, you have to basically bully a player whose character isn't even getting to do cool things. I'm not a fan of it, really, and I don't like big house rules, so my players simply don't die a lot.
I don't get this. Anytime a PC goes to 0 HP, they are potentially 2 rounds from death (Rolling a 1 on a death save counts as 2 failures, one more and you are dead). Any attack against a PC at 0 HP is an automatic death save failure, or 2 failures if it is a critical (and if they are within 5 feet, it's an auto-crit).
It's really easy for PC's to die in 5e. Even with a Periapt of wound closure - stabilizing doesn't mean your failed death saves go away. Really, the periapt just means it takes an extra stab to be sure.
What makes it hard to kill PC's is the presence of allies with ranged healing magic, healing kits, healing potions, etc. Even though it's easy for a PC to die, it's even easier to save them, if the party is motivated to do so.