This is an old Roper, I'd guess close to 30 years old based on the serial number (which would probably date it to 1982).
The switch isn't a kill switch like a deadman, its just a key switch to turn it on and off and is located on the carb shield housing. This isn't a new snowblower by any stretch of the imagination. Anyway what the deal was is it appears to work by grounding in some fashion, when its on the on position I think it grounds back or completes the circuit somehow. So what they did was take and run a wire off the on side of the switch to back to where the plastic grommet connects through a whole in a metal piece which mounts near the engine block or something near it via a tapped block. This effectively bypasses the switch but the same thing could be done by grounding the carb housing back to the same spot or removing the grommet and bolting the wires to the housing instead of making it a loop which is what the switch effectively does. What appears to happen when the switch is "on" it completes the loop.
What happened was the terminal, which was just a push terminal, where the key housing was corroded (it was thin aluminum, or something similar) and the vibration over time broke the connection, and poof the engine won't run.
Anyway after figuring out what they did I just grounded the things together and it runs fine, same as pushing the end of the terminal onto the carb shield housing.
You don't really need the switch to kill it, the switch on the carb housing it is inconvenient enough as it is, its an old pre EPA device the range of the throttle is enough to kill it if you put it all the way down. I can't see the logical reason for the kill switch unless you wanted to prevent someone without a key from running it, and even then it is bypassable.
Whatever the deal is it runs fine now. Rules out a lot of questions as to why it would stop running suddenly.
If you're interested in a bit of background... The history of this thing was I garbage picked it by my girflriend's (now my wife's) house five years ago because it had a huge "FREE" sign on it, with two flat tires and the wheel drive didn't work though it was running. I actually had to push it quite a distance down the sidewalk because she refused to help.
It sat outside her house for a couple of months because I had a truck which I drove only in the winter because it had no AC and poor ventilation during the summer. I picked it up in the spring and finally had her dad help me load it and hauled it to my house.
Aside from the flat tires they had replaced the friction drive wheel but in error forgot to put the chain back on properly so I reconnected the chain and fixed the drive wheels. The auger belt was toast, so I put on a new belt (using some $3 cheapy which in retrospect I probably should have just used a good one from the get go). I filled the tires with air. It ran so we tried using it come winter. My dad had the big tractor snowblower but for lighter snow it was always inconvenient to use the big one.
The problem was it had no power. It idled too low when it was cold and had no power. It would run but barely threw the snow. So in the spring we loaded it in the truck and took it to my Grandpa's house and put it in the barn. I was always convinced it only needed some carb work but never had the time.
This summer I decided to try and get it useable. My wife's old Troy Bilt's pull starter broke off last winter so I just shortened the rope and tied it off in a pinch, since it has no electric start with the short stroke was hard to start it but it got the job done. I was convinced I could fix this thing. Plus the little 5hp Troy wasn't up to the task of a large amount of snow. Clearing the end of th driveway after a heavy snowfall was a major chore for it. So I figured I'd fix this, sell it and buy one for my tractor. We had to push it up the loading ramps to the truck though because the tires were flat and it had no gas. I took it back to my house. Since I had drained the gas it actually started when refilled after replacing part of the gas line. I disassembled the carb, cleaned it out, replaced the o-ring, put it back together and adjusted it until it ran. It ran good, had great compression, burned no oil. The belt, which was a cheapy wasn't so hot by this point I replaced it with a heavy duty one and slapped it back together and tried to sell it.
My street has moderate to low traffic at best, the fall was warm. I'd be driving it out to the street and chaining it to a tree on my days off. No one seemed interested. The weather got colder and they started forecasting lake effect so I'm like "the heck with it, if no one wants it I'll just use it myself" because I was convinced I had this thing running good. I'd by then have to go buy the snowblower for my tractor, bring it home, unload it from the truck remove the lawnmower, install the snowblower which was a lot of work for a short period of time. I even gave the thing the nickname of "The Beast". At the least at the time it saved me $1000 because I never bought a snowblower for my tractor and technically saved me the time of going to get it and installing it.
So I lined it up side by side with the Troy Bilt snowblower in the garage and waited. I started it a couple of times because I was worried about the gas going stale (as I start my generator every couple of weeks for a half an hour or so for the same reason). Has snowed all of once in any amount you'd actually consider blowing it. We got maybe 4 inches of fairly heavy stuff (more than you'd like to shovel by hand) which I used to tune the under load idle, for which the thing ran fine and threw what was some pretty heavy snow. Parked it in the garage again. About a week and a half later later they start forecasting this huge storm (which never materialized) so I took it out before it hit and started it to make sure it would start, after getting to the point where it would normally be warm it died which is where the above happened.
I don't mind tinkering with these things. I don't really trust the Chinese snowblower engines quite yet (this is maybe the second really solid year of them on the market). I was worried that either the magneto or points/condesner (never pulled it apart to that degree or researched what it used) was toast, which would require major teardown and if I got into that luck would have it there would have been heavy snowfall. Of course in the end it turned out to be a minor issue and there is no snow for Christmas. I guess it all works out well in the end.