I think we tend to forget that the whole concept of "stopping power" and the obsession with terminal ballistics didn't really exist in earlier times and didn't enter the general debate among shooters (and that would include police officers) until after WW II. In terms of personal weapons, we tend to focus on the famous big guns, but even during the single action cartridge era, the actual number of Colt SAAs and S&W SA .44 and .45 revolvers was dwarfed by the gazillions of cheap, usually top-break .22, .32, and .38 pocket revolvers being churned out by countless manufacturers, which indicates there was demand and people found them adequate. Study historical photos, even of frontier towns, and try to find anyone who seems to carry a substantial handgun; movie cliche nonwithstanding, a little pocket pistol in a vest pocket was much more likely than a big .45 dangling from the belt. Even during the black powder era, by far the most widely produced Colt was not one of the .44's or even .36's, but the little Colt Pocket 1849 in .31.
Add to that the fact that most urban officers, and by the 1890s they made up the majority of American lawmen, did not carry open holsters and had to stow their gun somewhere under a stiff wool frock coat, and something compact and lightweight was very welcome.
So I don't think any officer would have seen the adoption of a .32 revolver in 1896 as a step down; except when riding with a posse into the wilderness, most officers of the law probably had been carrying something along those lines already.