Lanefan
Victoria Rules
I'm nastier as a DM and thus much prefer to get it over with. Even a three-save system could be concatenated into rolling a single save with double-advantage (i.e. roll 3d20 and take the best).That is how 5e uses saving throws in most cases... Duration is fixed and you can escape depending on your ability to withstand.
And actually this is how saving throws in 5.5 should be used.
Effects that kill/disable with a single saving throw actually don't fit into 5e. They should all be changed to at least 2 or 3 saving throws. Death saves as you say is a good idea, as well as stone to flesh.
Another factor is how difficult those saves are to make. On a single-save system, if you only need to roll a 4 to succeed your odds are pretty good. But on a 3-save system where you need to roll a 17 the odds aren't great even though you get three tries at it.
I like this except for the last line. Spells like polymorph should all be fire-and-forget, such that after casting it on one foe I can cast it again on another foe next round, meanwhile my first victim is still making saves.1st saving throw: minor effect, second: major effect: third: potential kill.
Dominate monster or polymorph could easily work that way.
Dominate:
1st: charmed
2nd: confused
3rd: dominated
Polymorph:
First: incapacitated
Second: stunned
Third: polymorphed
And those spell should at least require concentration or even a bonus action to keep up until the full effect takes place.
Dominate's a different beast; in that after dominating someone you then have to spend at least some effort in telling it what to do. Here the three-round idea makes sense, even if not the three-saves.