Cops in plainclothes simply do not openly display their firearms without any identification as officers.
Perhaps where you live cops are all in favor of open carry and do so while off duty. Most places, they'd be lucky not to end up with a suspension.
Last month a very tall and distinguished-looking gentleman walked into my nursing home (actually the nursing home where I am pulling an on-going private duty contract, performing one-on-one care for just one client who is a resident there) wearing a very nice business outfit WITHOUT the suit jacket. Nice dress pants, nice white dress shirt, nice tie, killer men's dress shoes, impeccable haircut, and frankly the guy had a very happening physique underneath those super nice clothes.
And he wore a sidearm on his belt. But no badge.
He came into the dining room with a smile on his face and a gun on his belt, then sat down for lunch across the dining table from me, chatting with an elderly rehab patent named Billy and calling the old man "Dad." Nice chit chat between father and son. (As a minor set of details, he was obviously a left-handed individual since his gun was on his left hip, and it looked to my eye like a Ruger.)
Three days in a row, he came and sat down with his Dad for lunch right across the table from me and my client. Three days in a row, a gun on his belt, but no badge.
On the fourth day he was wearing a badge visibly on his belt. I tried to read the badge to figure out if he was a private security guard or a municipal cop. (My eyesight isn't good enough anymore to read a badge from 10 feet away.)
I asked a nurse later on who told me Billy's son was with some division or branch or whatever of the Mass State Police. (I didn't want to find out his professional affiliation so I could report him or get him inn trouble, I merely wanted to know who the dude with the gun was.)
Why he didn't have his badge three days in a row I do not know. I myself (by law) am supposed to be wearing my own badge every day I am on duty, and yet every now and again I forget it at home.
Sorry to disagree. Perhaps not all who are theoretically in favor of open carry being legal are idiots, but pretty much the ones who actually do it are.
I can't speak to that since I know no one personally with an open carry license.
::ETA::
Maybe the badge was in the pocket of his (endlessly absent) suit jacket, and maybe he always left the jacket behind in his vehicle out in the parking lot.