The one thing that I just realized that I don't like about this system is that it's impossible for a character to die the moment he's hit. In most situations, the character will live for a few rounds after dropping below zero hit points.
It's possible - the moment a creature falls to negative hit points, it must save. Regardless of fort modifier, rolling a 1 means instant death. In practice, most characters are felled by strikes that don't bring them very far below 0; for them only a natural 1 matters. That's intentional; a normal strike should only with unusual luck be able to instantly kill.
Also; don't underestimate the risk of rolling a natural one. Since you must roll each round, even though each roll is fairly likely to succeed, it's only a matter of time until that 1 comes up. It's 78% likely that
even on minor damage you'll bleed out on or before the fifth roll. A character which drops frequently - even if he's healed right away - will eventually roll that 1; Mathematically, the chances are not insignificant, and anecdotally, I've seen it happen (this was, of course, right after the player smirked that he could only fail on a natural 1 anyhow...). Be sure to point this out, incidentally, to players that feel it's quite a safe bet; many people don't realize that it's quite likely you'll eventually roll that 1.
If you find the rolls too easy, the basic system is easy to adapt; just raise the DC of the Fort save. On the other hand,
You wrote, "What we need are rules that (normally) make dying a nice and slow process again." I'm not sure that that's exactly what I'm looking for. Instead, I'm looking for a system that makes dying potentially either a quickly lethal process or a slow process. I want the chance that a character will die immediately when dropped below zero hit points and I also want the chance that the character could take a while to die after dropping below zero. And, I don't want to know with 100% certainty, before-hand, which will be the outcome.
That's basically what it's designed to do as written. If the creatures damaged by anything vaguely threatening (so that you need just a 2, say rather than a 1) then the chance of death jumps to 2/3 - and that's if you need just a 2 to save.
The idea is that the system makes it very likely that a significant hit will kill you. Insignificant hits (when you only need anything but a 1 five times in a row) usually don't kill you, but even those aren't very safe bets. On the other hand, it's not that likely that a creature will die instantly, and no one knows when it'll fail.
You wrote that with this system, "At best, you still have a 1 in 4 chance of dying if unaided, even if you can only fail on a natural 1." What changes would need to be made to make this a 50% chance of dying while maintaining the simplicity that you've achieved?
Well, any hits that bring your hit point total down to lower than twice your fort mod the system already makes survival exceedingly unlikely. If you want to make natural stabilization even less likely, the easiest fix is just to remove it; Don't make the death saves stop after the 5th save, but let them continue indefinitely (100% chance of death, eventually).
Supposing you then find that too lethal, to then get a 50% of death, let a character stabilize on a natural 20; then you're basically letting the character flip a coin: will he roll a 1 or a 20 first?
Personally, if I thought the basic system were not lethal enough, I'd just remove natural stabilization entirely. Stabilizing on a 20 has it's downsides; namely that a character might stabilize right away, and that's just boring (IMHO). We've often (not always) played with the saves vs. dying rolled open on the table; it's a great tense moment and when it happens it's a constant reminder to everyone that their friend is down - but if you do roll openly, and that friend rolls a 20, you know he's safe and can safely delay healing him for a round. And avoiding
that kind of behavior was the initial inspiration to make this system in the first place: I never, ever want a dying character to feel
safe.
Recap, your options are realistically: keep it as is (minor wounds have only a 22% mortality rate), make stabilization take more than five rounds, remove stabilization entirely (100% mortality rate, but minor wounds can take long to kill), or finally, stabilize only on a natural 20. I'd pick one of the first three options and tell your players you're still tweaking the number of rounds.