I don't think you need to create a death spiral. Just the attitude that if you go down, you're probably not getting back up unless the rest of the group can survive the encounter and give you some out-of-combat healing. Staying up becomes the challenge of a combat as much as defeating the opponents.
The problem I see here is that the actual effect that a lot of people seem to want is mostly readily provided by means that these same people seem to reject.
To wit, if you want the attitude that going down is bad, then if nothing else, a character going down is part of a "party death spiral". Moreover, it's most noticable in smaller groups. (One of the reasons that I get away with so much grit in an otherwise epic game is that I run routinely for 6+ players, where losing a character temporarily is not nearly as big a hit as a standard party.)
So what people really want is a death spiral that can be put off for some time, so that the threat is always there, but not readily felt in most fights. If we were looking at some crude measurement like penalties to attacks and saves (from wounds or whatever), then instead of a straight -1, -2, -3, etc. from damage or wounds, you want something more like -0, -0, -0, -1, -1, -1, -2, -2, -3, -4, -5, etc. Maybe the first time you hit zero hit points, you pick up a minor wound on the track, and this puts you at -0 for rolls, but that much closer to getting a real penalty.
Functionally, that is way too crude and counter-intuitive. Moreover, it is too deterministic for a lot of play--you can't get the balance right to make it a real threat for some people without making it too strong for other people. Thus the obvious answer of "hero points" or such to replace the -0 penalty portion of the chart--or in other cases, DM fudging to essentially do the same thing. This time, even though you got knocked out, you don't take the -1 of the real penalty. The advantages of both fudge and "hero points" is that they can be gained, lost, and recovered on a completely different schedule than hit points.
Hit points are a pacing mechanism. If you want to square the circle of real threat in the instance with heroic or gritty struggle against threat in the long-term, then there
must be at least one additional pacing mechanism besides hit points. That second pacing mechanism may not be sufficient, but it is necessary.