Going back to the original question...
Should cops learn BJJ?
Nope.
Nor should they learn Taekwondo, Bando, Bujinkan Budo Taijutsu, Karate, Aikido, Krav Maga, or any other specific martial art.
Police Control Tactics or Defensive Tactics are designed around their job and their mission. They balance training time, and liability with use of force laws, officer safety, and the likelihood of controlling a subject without injuring them. They tend to be selected and pulled from various martial arts, with different styles coming in vogue at various times, and depending on the interests of various lead instructors. In the best programs, they have a unifying set of principles that are consistent throughout. In some others... they aren't.
At various times, programs like Pressure Point Control Tactics (PPCT), KMWW Force Protection Program (and other Krav based models), Gracie Combatives (and programs from BJJ splits), and other stuff that I can't think of get popular. Very few are really designed specifically for LE needs -- though KMWW and PPCT are. The advantages of packaged programs are documentable curriculums, instructor certification qualifications, and "name recognition" when presented to the brass. But they may not meet a specific agency's needs. For example, there's a guy in Alaska named Steve Jimerfield who -- based on his professional experience as a trooper there, as well as high ranking black belts -- designed a program that takes into account the layers of clothing worn by officers and everyone else up there...
With all that said...
There's no reason why a cop shouldn't train in BJJ, Filipino Martial Arts, Bando, Krav Maga, Judo, Karate, Aikido, Tai Chi, Bagua, or any other martial art. Trained with an eye towards function, and used within the use of force models, any of them are beneficial.
I will note that, increasingly, there are BJJ schools that are highly sport oriented. Also, getting someone in a submission hold is fine -- but you have to be able to cuff them, and that takes an appropriate emphasis in your training.
And I do respect a couple of things I've seen some of the Gracies do teaching their program. When cops called them on a technique, saying that you can't move that way with a gun belt on -- they tried it, and then adjusted to work. And they've actually tested grabbing the slide in a gun disarm... which I admit, I haven't!