Get a Grip
I ran across this article about grip strength a few weeks ago. It's an interesting piece. The gist of it is, grip strength has long been associated with robust health and that modern culture has contributed to not only our hand strength withering away, but actually altering the anatomy responsible for good hand strength.
Our arboreal ancestors needed grip strength to swing through the trees to hunt, forage and evade predators. Infant apes can hang onto their mothers for extraordinary periods of time because to survive they have to: evolution has selected for high grip strength. Even human babies have excellent grip strength.
One of my former athletes is a world class obstacle course racer. Her first year of racing she tore up the pro ranks because of her two superpowers: her grip strength/upper body strength (from rock climbing) and her elite distance running speed. If you fall off the monkey bars, or can't pull yourself over obstacles, you lose time and get penalties.
I suspect, though I have only my own observations to back this up, is that a lack of grip inhibits other expressions of strength. An interesting test of this is the Rolling Thunder dead lift handle that Iron Mind sells.
What I've found is, that when you reach the limit of your grip strength on this tool, your legs won't work, even though they may be more than strong enough. It's a bizarre sensation. This is why, I believe, that straps allow you to lift a heavier barbell than with a basic overhand grip: the grip is taken out of the feedback loop. If your nervous system senses the grip isn't up to the task, it will inhibit the effort.
I used to say that improved grip strength is free improved strength everywhere.Those 1RM deadlift tests where you reach your limit and the bar stays glued to the floor? Next time that happens, grab a pair of straps and see if it comes up. My bet is it is your grip holding you back.
Training your grip won't necessarily improve your health. It's only associated with good health but is not causative. However, training your grip might improve the quality of your everyday life and it might improve your overall strength.