Yes, I have three kitchen timers I have confiscated off fences.
This was a decision made several years ago by our then Seattle Tacoma UIC and is still our steadfast policy.
Part of the reason is that we are in control of the game. As SRW pointed out, our coaches know that game time starts with the first pitch (unless the home team REALLY dawdles, in which case we simply loudly announce that the clock has started, which usually gets them racing out onto the field). If we have someone in an invitational who is not from our area, we kindly explain to them time starts with the opening pitch and please do not dawdle. We expect the home team to take the field ASAP after the plate meeting.
We usually do not shave time from games, but not saying that hasn't been done. Heck, as often as not a coach who is getting his/her butt handed to them will say something to the effect of "Hey Blue, aren't we done?" in which case we will usually agree with them. With a timer on the fence, you eliminate that option. Often, coaches do want to play the complete game or at least until time expires, but sometimes they do realize enough is enough. However, we are expected to NOT shave time from a game on our own.
We have not encountered a problem with coaches keeping UNOFFICIAL time. About 89% of the folks up here have cell phones, and last I checked they all indicated the time. When we have the official timepiece in a pocket or otherwise on our person, we have more complete control of the game. Plus, IMHO, there is nothing more annoying than that timer beeping back there on the fence -- usually starting about mid-pitch and throwing everything off kilter. Umpire has to call time, go back and stop the beeping, yada yada.
If it is your chapter's/TD/UIC opinion that you have a timer on the fence, then by all means use a timer on the fence. Just don't bring one to Seattle Tacoma, k? Otherwise, it clutters up my trunk and have to find that damned beeping timer when I throw something in there and it hits a button.
