I generally wear a western style vest for concealed carry, works well with my watches..I have a few pocket watches but seldom carry them. Most non-jeans pants do not have a watch pocket and the jeans watch pockets are usually too small. And then there is the problem of where the chain is to be attached along with it interferes with carrying a pistol, especially pocket carry.
As to wrist watches I took mine off when I retired, only wear one for specific reasons. Mainly rely on phone or clock in vehicle.
An interesting thread as I live in Elgin of watch building fame. Recently a parcel of land here was totally fenced off to allow remediation of the radioactive material on site. An interesting story. Luminescence Disfigured and Killed 1920s Watch Painters - ALEXANDER
I initially wondered if this was an overreaction until I read this: Radium-226, used in vintage watch dials, has a half-life of approximately 1,600 years, meaning these dials remain essentially radioactive for centuries, even if they no longer glow.
I remember my grandfather having one when I was young. But the only time I remember him carrying it was on Sundays when he was in his suit and dressed for church. He was a farmer and I guess he didn't want to take a chance on damaging it while working on the farm.