Difficult to start a four stroke mower when cold - how to diagnose the problem
I had an identical problem as yours with my Toro 4-cycle mower. Eventually, the mower wouldn't start at all, which I tracked down to a dead ignition module (ignitor). After I replaced it, it's been easily starting for about 4 or 5 years now. The ignitor looks like this:

This may or may not be your problem, but if you've eliminated everything else...I used a piece of card stock to set the gap between the ignitor and the rotor.