A shot putter would train to be strong and powerful in their whole body.

Not doing the same exercise over and over stops you "maxing out". That's why sprinters do heavy slow squats, light fast squats, jump squats, box squats, wide squats, single leg squats, lunges, side lunges, jump lunges etc. not to mention the speed work they do on the track.

Push-ups work max strength if your really weak but if your normal they're for muscular endurance or can be for training power.

Studies have shown that periodising your training by varying speed and intensity works best.

I wouldn't 1rm your cable thing, it'll mess up your biomechanics. 3/4 rep 95% bench press then transfer the strength gain into power.