I would like to think that a front suspension on a police bike is only there to cushion the jelly belly roll of our doughnut eating aficionados on bicycles while going off curbs. But on a less or more serious note, depending on your perspective I see here in our city that it is less functional than it is political.
In Our City
In our city in the USA (politics may work differently in your country or city) our 'mounted' police that ride police bikes do not do so on a daily basis. They are NOT bike police. They are regular cops that drive cruisers and during downtown special events, a couple of them get out the bikes to facilitate more approach-ability and ease of movement through crowds from one end of downtown to the other. Of course our city had just purchased a couple police Segways. So they normally get the Segways out as they use less energy and burn fewer doughnuts. But I have seen the police bikes at time strapped to a police SUV bike rack and being taken somewhere. Maybe it was to the repair shop who knows as they rarely use them to chase perpetrators through wooded areas and trails as they have dogs for that purpose.
In our city we have few stairs and I couldn't imagine our officers taking them down stairs as they are no where near proficient or capable of navigating a bicycle around technical terrain as they only use them sparingly during special events. Our city police force of 130-200 (depending on year and budgetary constraints and layoffs) has a peddle pool of only about 10 units.
How it normally works and why they are the bike they are
In our city here, the city council approves a budget for every city expenditure. I have sat in many city council meetings and watched these approvals. They may approve $15,000usd for 10 bikes putting them in the $1,500 range. They approve the expenditure amount for purchasing new bikes usually to replace or purchase them all at the same time so that they are all the same type, year and model. They have an open bid time period where they may except open bids from local bike shops to other shops not local and online bike suppliers that may or may not have have specific 'Police bikes' for the purchase contract. They provide a very basic requirement list which usually has less to do with the way they may be used and more to do with equipment that needs to be mounted.
The bike suppliers providing bids to this contract and rarely if ever provide any recommendations as to use or bike specifics and features. If a bike in the $1,500 range comes with a suspension fork, removing it and replacing it in terms of adding an additional fork (ie -$300 fork removed, +150 for non suspension fork, labor $50 = net savings $100 per bike) and labor would almost be a wash and not worth the time for them to consider. They accept a bid and contract is fulfilled and closed.
Our city officials or police officers that use the bikes are not technically literate or knowledgeable enough to know what they want or need (though some cities in the world may have very capable and knowledgeable cyclist among their ranks) and unless they are directed by the contract bidders to an alternative, they will get a typical run of the mill police bicycle that is good and versatile for ALL terrain, obstacles and functions for all and every city. And that is the way it works.
We normal or avid cyclist
Now we, more normal, avid, every day cyclist may see a police bike as not specific enough to their needs and thus these bicycles, in many cities and in many terms, are not as specific as they could or maybe should be. Maybe a cross-bike with straight bars and a cross tire tread would be better suited for police work via bicycle. We, the ones that ride hundreds of miles a week on various types of bicycles are more averse at being able to determine bicycle use requirements in terms of bicycle type. But then we are not (in most cases) normally involved in budget approval, contract bidding and fulfillment.