I've been really thinking about this question since you posted it, but it has only lead me to more questions of my own!
Jesus, during His ministry, saved many people from physical death, illness, etc. But I think the big thing about Him is that He saves our souls, eternally.
I do not at all doubt God having a role in saving people's lives when it is not "their time to go," but couldn't that be One of Jesus's triune Partners, the Father or the Holy Spirit? I guess it could really be any One of Them; They certainly don't have a set of rules to follow!
Miracles happen and people cheat death all the time; regardless of whether God saves them in these cases, it is probably a good idea for them to THANK Him for letting them stay awhile longer.
My wife's young cousin was killed in a car wreck a mile or so from his home, a head-on collision on a neighborhood road. A month later, my wife was in a serious collision with an 18 wheeler on an interstate highway, and walked away from it. Her cousin was driving a Mustang convertible and hit a similar sized car, and my wife was driving my SUV that has a frame all-around, and after being pushed sideways 300 yards at 70 mph, it was repaired and I still drive it today. Although I think that God was present with both of them in their respective wrecks, I think that the nature of their accidents and vehicles played a role in their fate. But if my wife and I were to say that God saved her, it is kind of awkward because of what happened to her cousin; do you catch my drift?
A very good and thought-provoking thread though, as yours usually are.
~U2Alabama