I fondly remember the stick days. My Olds had the 3 on the tree Saginaw. It would stick in 3rd if you didn't go completely up into the neutral gate before pulling to go down into 1st. I would have to kill the engine, set the brake, and get under the hood to pull up on the linkage on the steering column. Another time, my clutch pivot broke and I had to shift without it and was fine until the red light. Then I put in a wide ratio Muncie 4 speed and a Hurst shifter. The Muncie had a bad 4th gear synchro and would kick into neutral if I backed off without holding it in.
The winter driving is where they really shine though. Back then, before the traction control systems and ABS, the precise control of power to the rear wheels could really save the day.
But now I live in Chicago and this city never needs an excuse for a traffic jam. It gets pretty old to spend 75% of your time holding the clutch in. So my last 2 cars have been automatics.
My funniest memory of a manual is from college days when I had been driving my Olds with the mechanical linkage and racing clutch. I got into my brother's 240Z with the hydraulic clutch. I thought my left foot was going to go through the floor the first time I hit the clutch. My best friend used to challenge me to get my car started w/o the starter on all but the most level lots. If I could get the car to roll at all, I could get it to start by easing out the clutch in 2nd gear. That came in handy a couple of times.
As pilots, examiners can ask us to demonstrate competent use of any equipment on the airplane. I think it is a good idea for cars too. Any new driver I work with will have brought a car to a complete stop with both their parking brake and engine/tranny with a stick. I've never asked anyone to slam an auto into park though. They'll also learn how to shift without a clutch except from a stop.
As for distractions causing accidents or near misses, it was an open ice tea container tipping over that almost caused me to run into my cousin's car. He wondered why I stopped about an inch behind him. In the airplane, tuning more than 2 dials on the transponder between instrument scans, just about guaranteed a slow right turn. Destractions can come from anywhere.
Joe