An issue I've come across on several occasions with the later Briggs and Stratton auto choke system. This is on the later engines fitted to the new Hayter Harrier mowers, and other machines. I just happened to have four of these over a period of two weeks.
This engine uses the thermal device found on many of the Briggs and Stratton engines nowadays. Here's roughly how it works.
When the engine starts, and the governor cuts in, that pulls the throttle down so the engine revs at a safe and comfortable speed. This action on the throttle shaft also nudges the choke off enough to keep the engine running comfortably.
As the engine heats up, the thermal device pushes a little plastic arm, which then leans on the choke shaft to open it fully, and continues to hold it open while the engine is hot, so it's ready for a hot restart.
What's happening is when the engine cools down, and the thermal device backs away, but the choke shaft is getting stuck, and therefore leaving the choke in the off position. This leaves you with a engine that won't start when it's cold, or at least it will be extremely difficult to start.
The cause: The arm on the top of the choke shaft is binding against the carburettor mounting bracket. Now They should really press a dimple in the bracket so it clears the arm, or make the arm a few thou shorter. I have got around this by shaving a very tiny amount of plastic off the end of the arm, or even a few passes with a bit of emery paper is enough to cure it. Hopefully Briggs and Stratton will address this issue soon, if they haven't already.
Anyway, thought I'd share my findings. Check out the video below, it'll help to see what I'm talking about. This was one I had on the bench recently.
Edit: Just want to add, that even if your engine suffers from this problem, it might not happen every time you go to start it, or when you look and check for the issue. What I'm saying is, by the time you've trundled the machine from your shed to the lawn the vibration might have just been enough to release the choke and it'll start. So you'll need to actually check it as shown in the video.