Hiya!
I should have clarified.
This is what we did when we first started 5e.
About a year and a half ago...I "5e'ified" the whole idea of "as you die more often, you have less and less chance of survival and more and more a chance of never being able to be raised again" (basically, the whole reason for the -1 Con and ResSurvival Chance).
..
My current house rule is simple: When you get Raised/Resurrected, you immediately make a single "Death Save" (as if you had gone down to 0 hp). If you succeed, you are back to life, easy peasy.

If you FAIL, you are back to life, but you PERMANENTLY FILL IN one of your "Death Save" circles. This means you have only 2 'saves', with one being an automatic failure (as it is filled in already).
..
In a nutshell, eventually a person is going to fail three times when Raised/Resurrected, and that fills in all 3 Death Save circles...meaning they will never make a Death Save. Meaning as soon as they hit 0 hp, they're dead. Period. Forever.
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While the 1e/HM style worked fine, the "simplified" version I came up with a year and a half or so ago is more "5e Feeling" and works great (maybe it was a bit longer ago at this point?...I have no idea...time gets all wonky when you have no job and sleep at completely random times over years and years...and age doesn't help either!

).