Why did the .38 special model 10 become a standard police revolver?

The thinking at the time was that .45 was too big and had too much recoil for general police use. Less powerful cartridges were underpowered and considered unreliable for stopping a dangerous criminal. .38 Special was "just right".

Note that ballistics of the .38 Special cartridge are not too different from the 9mm Parabellum, which is in widespread use as a police and military cartridge today.

As for the Model 10, it was the very first gun chambered for the cartridge back in 1899, and was intended as a "military and police" weapon. Blackpowder Colts in .38 and .45 were in common use at the time. It didn't hurt that the Model 10 was the first "modern" Smith and Wesson, with several features that are common on revolvers today, but were "the latest thing" at the time.

Adoption was slow at first, but by the 1920s, many police departments had seen the wisdom of moving from black powder to more modern cartridges. Any semiautomatic will have a more complicated "manual of arms" than a revolver, which is why they were not adopted for widespread police use until the late 20th century, when semiautomatics had become both simpler and easier to operate, and more reliable. So,

1. It "had" to be a revolver.

2. .38 Special was the "right" smokeless cartridge.

3. The Model 10 was the "right" M&P gun for .38 Special.

The answer is why not. The model 10 was affordable, simple to operate, and dependable. Everything a police department would look for in a sidearm.

they found it kills innocent colored ppl easier