Look at it from another angle---police forces choose MTBs, so why?
Id wager it is because they frequently don't succeed through speed, and are quick enough when speed is needed.
A strong rider on an MTB can definitely catch a pedestrian, and follow through much terrain .... but obviously a pedestrian can hop a fence and be gone. Cops have radios.
Most fooling-around cyclists---the kind of people who would be breaking the law stupidly in urban areas---are no match for most cycle cops in fitness. The cops work out regularly, and the few bike cops I knew took great pride in being fast---rather, quick. They aren't looking at one-hour chases---they are sprinters. Of course, the cop is carrying a ton of gear--weapons, first aid, bulletproof vest, radio---and so isn't designed for a long chase. See above remark regarding radios.
Further, bike cops are primarily about mobility, not speed. A cop in a car in traffic sees a crime, s/he has to park, get, out, and chase on foot because the criminals are usually on bikes or on foot and able to cut through crowds, across public spaces---places cars could not go. A bike cop can duck down an alley, sail off a two-foot drop, shoot through a three-foot gap between buildings, and be ahead of the runner--and in pursuit, can keep an eye on the suspect while using that radio I keep mentioning.
If long-distance speed were all that mattered, bike cops wouldn't bother with guns, radios, flak vests. If instant acceleration was all that mattered, they'd all ride ultra-light road bikes. But what matters is delivering the fully-equipped cop to a given location more quickly than s/he could get there on foot after abandoning the car. And abandoning the car is itself problematical, considering there is usually a small arsenal in the trunk.
Rigid MTBs look like good tools for the job when one considers the actual job, not some fantasy cop movie where bold cycling cops chase renegade bike messengers or highly athletic small-time drug dealers down carless urban streets (which only exist in movies.) (In real life those drug dealers don't go on long, hard training rides, they have BMX bikes if they ride, and they smoke a lot.)
Cops could have chosen Any bikes. They chose what would best help them do their jobs.