Requiring 3 saves to avoid death means we have the same SoD effect, with a lesser chance of "D" and some ability of the character to act in the interim. Let's assume we change Petrification from "one save or turned to stone" to "3 saves, one every second round - fail the first and you are slowed, the second and you are immobilized, and the third leaves you petrified/dead. What does this mean in-game?
To the PC, it means three chances to save instead of one. It's less likely three saves will be failed, and in the meantime the character can still act while slowed. So this is a lesser threat, but sooner or later all three saves fail and the character dies. Over time, PC's make a lot of die rolls, so one roll (however unlikely) that will have a huge impact will get rolled eventually.
What about to the wizard picking spells? Well, if he picks Stone to Flesh, the monster gets three chances to save, if he succeeds on the first he's slowed but still acting, and I have to wait until round three to see whether he will be immobilized, removing his threat. If combat is generally over in three rounds anyway, what good was that? Pick a spell more likely to work. This was the problem SoD/SoS/"save and nothing happens" spells in 1e/2e - they were too chancy, especially as saves improved at higher levels, to be worth it. Just hit him with another Magic Missile/Lightning Bolt/Cone of Cold - a least that does something every time.
I'm not a big fan of "Save or lose the character" effects. If character life is cheap, I think it does lead to players not investing in the characters. No sense spending more time on his personality, backstory, etc. just to have him killed his first time out - I'd like to spend more time playing the character than creating it. Higher death rate games, as noted above, need rapid character generation systems as well. Of course, that's back in the days of random character generation. Now, we get a lot more choice (point buy stats, feats, spell selection, etc.) If my tricked out fire wizard dies at 4th level, I guess I can always bring in an identical tricked out fire wizard to replace him. Now how impactful was that death? Plekor the Pyromaniac is dead - long live Felgar, Bringer of Flaming Death!