Beat cops don't like lanyards, or sam browne-style cross belts, or suspenders, or anything else that provides an easy handhold (including braids and ponytails on saavy female officers)because it will get grabbed, and you will get slung around by it like a sack of frigging potatoes. I could also see it regularly getting hung up in seat belts, MDTs, radios and all of the other gear in the front of the car. I would not want to use one on patrol.
As was already mentioned, the lanyard in military service began as a way for horse-mounted troops to not lose their sidearm when transitioning to saber. There is probably alot more to it than that, but that is the simple version.
In my day, it was used to keep from losing the darned boat anchor pistol in the woods, and everything else that was sensitive was on a lanyard or a "dummy" cord, which was usually a strand of 550 cord (nylon parachute line). In fact, the GI-issue lanyards get used far more often, in my experience at least, to secure crypto "fill" devices (stores and loads encryption schemes into radios and 'scramblers'), NVGs, GPS units, compasses and anything else considered a "sensitive item." Lose that stuff in the woods, and you ain't leaving until it turns up, and you can and will lose stripes, pay, beer time and access to the nicer points of enlisted life.
I still have a GI lanyard, but I use it to keep from losing my keys on tactical stuff. I stuff all of the excess in my front pocket with the keys. It was also handy for fishing the keys out under armor, vests, leg-drop holsters and the like.