The number of pushups a guy or gal can do is a fair proxy for general fitness. 50 pushups in 60 seconds is a fitness benchmark. (I can not do that yet.)
Pushups are a calisthenic exercise but they do build some strength in the core, chest, arms, and shoulders. I do them on my calisthenics days but you can do them every day.
Why use the handles? Because you can dip lower, and because they are kind to your wrists. I don't approve of declined pushups because of the wrist and shoulder stress.