I have a rather anciet MTD mower that I purchased in 1990 and it's doing teh exact same thing. However, the cause was pretty easy to diagnose, it's a cracked primer bulb for the carb. Fortunately, these are very cheap and just snap into place. If you going into it, hit a garden equipment repair dealer and get a new primer bulb and replace the old one while your at it.
BTW, it's not had a lot of maintenance and has never seen Stabil, all it gets is an oil change every 2 or 3 years. People may trash B&S for making crap but the service I've gotten from that 130 dollar mower has been exceptional. I'm not sure, but it may still have the original spark plug in it and it still starts on the first pull and still has good power.
I've also got a 12 hp Murray rider of nearly the same vintage with a B&W engine. That one has been a bit more of a challenge. The front axle has been welded twice because it's just stamped steel and I really should just make some bronze bushings for the kingpins, however it still drives straight enough despite the somewhat floppy front wheels so it's a why bother deal for me. I also now have an issue with an oval float valve seat on the carb, which is not replaceable and a new carb is 120 bucks minimum. Current fix is a fuel shutoff but every time I forget to turn the gas off it means another oil change. This year it's has been 4 oil changes because of a crankcase full of gas when I forgot, I'll bet the inside of the engine is clean as a whistle at this point. Other than that, the B&S engine has been flawless.