velojym

I helped a mid-sized city's bicycle officers train many weeks. Some of them developed into serious cyclists, but for a good part, it was a place for the PD to put officers who weren't meeting their physical requirements. More delicate road bikes would have been a maintenance problem for some of the less-experienced riders, and especially for the better riders using more... vigorous riding styles.
Most of the job was patrolling parks, downtown, and other areas where vehicular traffic was excluded or just so congested as to be useless. I did accompany them on longer rides, for fitness purposes, which they realized they weren't getting from just noodling around the park.

There's a LOT of curb hopping, crossing grass, etc. on a normal police bike, and the adapted mtb they chose did the job really well. Though they had a lighter loadout than most of the rest of the force, they still packed on enough gear that a road bike wouldn't be the greatest thing in the world for them, all of the above considered.

The greatest direct benefits (remember, for any 'real' and dangerous issues, they all had radios) for the PD were:
-They weren't driving around in a steel and glass cage. More accessible, and had to learn to interact with the public constantly. This helps to alleviate the 'rolling fortress' mentality.
-Officers who needed more physical activity, so as not to appear to be the stereotypical Uniformed Donut Muncher.
-Provide a Police Presence in areas where cars are either excluded or unwieldy, with a LOT more mobility than a foot patrol
-etc...

I have my own opinions about police practices and whether/how they enforce certain laws (not for this forum, however), but I did really like what they were doing with the bicycle program.
In my experience riding with these folks, I'm pretty sure trying to use road bikes would not have worked out well.

...and, for the record I'm a roadie.