I had this problem with a 18.5 hp Poulan Pro for years and it was very annoying in that: We are in the southeast and we want our grass cut before the heat comes. We had a professional law repairman to replace all of the things you named and the Poulan started running real good for 20 minutes and then we had better have a pushmower or swingblade around. I read someone's post and the post said the carburetor components were causing this problem. I went to Ebay and bidded on a new briggs and stratton carburetor for $57.00, and now that cut off problem is gone.
This may help 50% of lawn mower starting issues! Try adding a gasoline additive to boost octane and clean fuel lines. There is a product made especially for small engines that cleans the gunk formed quickly in lines and moving parts from current low emission fuels. Where did we think the stuff goes that doesn't burn and doesn't go into the air? - It stays in our mower's engine causing moving parts and plastic fuel lines to bond together!
After forgetting to empty the mower engine over winter, I found nothing worked to get it started. My local small engine guru suggested adding a fuel stabilizer, octane booster and fuel line cleaner made for small engines to fresh gasoline (read directions). Try to turn over a few times to get fuel moving into lines, leave sitting for a few hours then try again to start. It smoked and coughed the first few times, but gradually improved greatly, now starting much easier. And re-starting easily after being turned off! I will remember to winterize this year.
Another tip: keep the housing around the mower blades cleaned of dried on grass.
Also, if you are cutting grass with deck significantly lowered, try to start mower in an area with very little grass. Blades that rotate when you are pulling a starter rope are hindered from turning by contact with heavy, thick grass, preventing easy starting.
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