Something to watch out for with these engines is when they are like you mention Norm , the compression can be low, if the comp is low even if you get them to start well cold they can be very difficult to start hot if it's a compression issue.
I've had a lot of mowers that do the same thing ,start on starter fluid easily but harder to start without starter fluid ,most of them I throw them out as when I test the compression pressure with the comp tester on a hose the comp is under 50 psi, you loose around 15 psi with the comp tester having a hose on it. When I use these mowers to cut grass they lack a little power and stall a little easier in long grass ,then when the motor is hot and stalls in long grass they are a pain to start again ,some times taking a lot of goes on the starter to get going again ,so this is why I don't bother fixing these mower engines if under 50 psi (65psi with the stumpy comp tester),.
If you haven't tested the comp pressure or tested the mower when it's hot cutting grass it may be too worn out. I have one now that is the same as your one to start and I won't be fixing it as the comp is under 50psi.
