Doing push-ups prevents injuries

. . . push-ups can provide the strength and muscle memory to reach out and break a fall. When people fall forward, they typically reach out to catch themselves, ending in a move that mimics the push-up. The hands hit the ground, the wrists and arms absorb much of the impact, and the elbows bend slightly to reduce the force.

The article continues:

The push-up is the ultimate barometer of fitness. It tests the whole body, engaging muscle groups in the arms, chest, abdomen, hips and legs. It requires the body to be taut like a plank with toes and palms on the floor. The act of lifting and lowering one's entire weight is taxing even for the very fit.

While I agree with the first point, there are various ways to do push-ups. For example, I don't do push-ups with my "palms on the floor."  I only do knuckle push-ups.   That's because if I bend my writs it exacerbates a longstanding carpal-tunnel-type weakness there.  Variations of the standard stance can help you to train different muscle group: try doing push-ups with your hands by your waist, or hands close together under your chest.
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NY Times