Briggs & Stratton won't stay running; need advice

Greetings,

I'm trying to help my father repair his lawnmower, a Craftsman 21" high-wheel push mower with a 625 series Briggs & Stratton engine (4-cycle). The mower is Sears catalog # 917.388114. The manual & engine info is here: click

Hopefully we can save this one from the graveyard as cheaply as possible. I'd appreciate your advice on what things to do.

The problem is that the mower doesn't stay running. Dad says it'll start, but then the engine stops shortly after starting. The spark plug was replaced sometime last year, but unfortunately (please don't shoot the messenger) it's never had an oil change and he keeps old fuel in the mower, even over winter. I'm pretty sure the engine is need of a good cleanout, but I'm not sure what/how to do it.

My thoughts on things to try:

• change oil
• dump fuel tank
• change fuel filter
• change air filter
• inspect & possibly change muffler (?)
• pull spark plug & crank motor to clear out cylinder

Are there other things I've missed or worth trying? I'm particularly interested in methods for cleaning out the fuel system and carburetor components. Is there maybe an additive that I can put in the tank to do the job, or does it require disassembly?

Do you think this thing is gummed up beyond hope?

Thanks!!