The Winchester 140 is pretty much the same gun as the Winchester 1400.
These guns were cheap, and sold pretty well into the 1980's. They are a decent shotgun for the money and pretty reliable. My first auto shotgun was a 1400 in 12 gauge. I still have it, and I added a 20 gauge to the collection. I have always liked them. One problem with this gun is that they seldom bring much more than $150-$175 today.
I have repaired about 6 or 7 of these guns over the last few years, and the biggest problem is the plastic magazine throat breaking. I have seen 2 guns with the exact same problem as yours.
The screw that broke off is the only one in the bolt. It holds the cocking handle bridge onto the bolt body or carrier. When it breaks off, it allows the firing pin and spring to exit stage right.
Gun Parts Corp. has the bolt handle bridge and the bolt carrier body in stock...but I doubt you need either. I have had great luck drilling out the broken screw and tapping it for a larger dia. screw. I add a little lock-tite and put it back together. Never got these guns back again. For $60 you can buy a new bolt body and new screw. For the price of the screw and labor you can fix what you have.
Tell your gunsmith to fix it. I wouldnt get rid of a decent gun for $150...that won't buy you a new one.
Any questions on the repair...email me
buckmark@adelphia.net