Why navies independently adopted triple turrets

Did the U.S. Navy copy triple-gun battleship turrets from the Tegetthoff class?

No. Multiple navies pursued triples because one turret could fire the three-shell salvo used for long-range ranging and correction.

Several navies pursued triple turrets because a three-shell salvo was useful for long-range ranging. One turret could fire a salvo intended to straddle the target, and another could fire a second group to check the range. The theoretical advantage is emphasized because arranging three guns together does not itself guarantee accurate fire. At Jutland, range information could also pass along the British battle line through range clocks, allowing other ships to begin with the range already found by those ahead.