I’ve seen studies like this, and they usually separate men and women. Peripheral circulation and nervous system goes first, and that leads to loss of grip strength. I don’t think it has any direct connection to falls, it means your body is failing. If you have chronic carido-respiratory disease exercise my improve QoL a bit and extend life slightly, but no silver bullet. If you do hard exercise your entire life, you will probably live longer with a higher QoL.
What you've written there strongly implies - on two counts - that "grip strength" is merely a proxy for some other (unknown) factor - perhaps "general good health" (which, of course, would tend to stave off dying). Also that the "increased risk of death" is not necessarily directly related to injury due to falls (but rather to "lack of general good health").
I highly doubt, in other words, that an exercise regime that focused solely on grip strength - to the exclusion of all else - would have any significant life-extending effects. (Though it might slightly reduce deaths / injuries resulting from falls.)
I agree with your premise (I am inferring) that the study was using grip strength as a proxy for a multitude of other factors (which might include: high level of vigorous exercise; excellent genes; etc.).
This makes the basic implication - "Improve your grip strength and live longer!" - next to worthless. You could have your hands amputated, and - if you eat right and exercise regularly (things that would ordinarily also enhance grip strength) - you'll live longer.
Regards,