Thanks everyone for the recommendations. I’ve decided to try to first replace the worn gears using a kit I just ordered from Amazon. Apparently, stripped plastic gears are not an uncommon problem for these garage door openers. My Sears opener was made by Liftmaster, and just about every Liftmaster, Chamberlain, Craftsman, Sears, Master Mechanic, and Wayne Dalton opener made since 1984 uses these gears. If they’re not thoroughly lubricated with a good low-temperature lithium grease, they will shred apart. I imagine the grease on my 1980s-era opener finally wore off, and that let the gear self destruct. It shreds the gear into tiny particles that look like snow.
The process for replacing the gear involves taking about half of the opener apart, and I’ll have to take it off the ceiling because I don’t have enough clearance to take the old gear out of the top. I figure there’s a risk that I could drop or break the darn thing, but I also figure that wouldn’t put me in any worse situation than I’m in now, and it does carry the possibility that I can fix this with $30 in parts and an afternoon’s worth of my time rather than spending hundreds of bucks to get a new one installed.
I do recommend that if you own a garage-door opener made by one of those companies above and you know it hasn’t been touched in years, you might want to open up the case and make sure the big white plastic gear has a healthy coating of grease. A cheap tube of lithium grease and a few minutes out of a Saturday afternoon could save a lot of money/hassle in the future.