Gerry, the PTO has a "safety" brake that stops pulley rotation when the PTO is turned off. If the PTO armature gap is too tight, it will engage without fully moving from the brake. If the gap is too open, the PTO won't engage reliably or not at all, particularly when hot.
When adjusting the gap, I use an .017 "go" and an .019 "no go" feeler. The .017 feeler should slide in freely. If you have to adjust one or more points, do a recheck of all three until you can walk around and feel the proper gap without adjusting anything.
On my 318, once the engine is warmed up (takes several minutes), I can engage the PTO with a 50" mower at any throttle setting between low and high idle. In practice, I generally rev the engine up a few hundred RPM from low idle, then flip the PTO switch. With the engine warmed, I can move the throttle to full throttle in about one second, and the engine will speed up smoothly as I do it. If I get a flat spot in acceleration, or misfiring, either the engine is not warmed or the main mixture is too lean.
I'll note here that my carburetor has an adjustable main jet in place of the standard fixed jet. So that acceleration test verifies that the main mixture adjustment is set correctly.
I'm at 5000 ft. altitude, so have no real idea how the fixed jet setup behaves at sea level.
The most common lean carburetion issues with an Onan engine are a plugged idle jet and air leaks in the intake manifold. Those are regularly discussed in postings here.