Well, I guess that is really the crux of the "issue" I am raising. As has been pointed out, using the bridge is a skill (which takes time to learn). It seems that shooting off-handed is another skill, quite valuable I might add, and I'm just surprised that people would rather take the time to learn how to use the bridge, rather than simply taking the time to shoot off-handed. With the ability to shoot well "either way", the number of shots that truly "require" a bridge are few and far between (snooker tables aside).
In the case of that 3 ball - shooting it left-handed is a trivial shot that causes no loss of "rhythm" - you plunk it in and move on. Getting out a bridge and lining it up, and hoping you don't accidentally bump something with it, making the shot, then putting the bridge away....just seems like such a big production and unnecessary interruption to one's game.
And that settles it. You're no good with the bridge. Neil was right all along. That's why you don't like it. It takes all of five seconds to pick up the bridge and get yourself aligned with it, and it takes all of two more seconds to put it back under the table. The real loss of "rhythm," as you put it, is missing that three ball, which I would almost surely do left handed.