I had one of those, with almost the same problems! Here's how i solved the problem: I backed my truck up to a hillside of my front lawn, carefully pushed that durned mower into the truck, drove to the local scrap yard, and made it a present to them, free and clear! Problem solved!
By: Rusty J. who hates Lawn boys!

Well, the problem I have with YOUR solution, is that I can't replace it with anything as light in weight, and I have to lift the mower up over retaining walls, and up steps, and over ledges to get to the yards I mow. Nothing I own is on flat land. No modern 4-cycle mower is light enough for me to handle. I have to keep my old Lawn-Boys running.

Wow, well...not sure where to start with that. You mention its hard to start....hows the plug look? Is it the proper plug? CJ14 or equivalent set at 0.035"?

Did you set the gap on the newer coil at 0.010" between the flywheel and coil?

Is there any debris caught in the cooling fins and in the shroud causing undue heat?

Any string wrapped around the crank/blade area?

Check to see if your primer plunger is working.
I had that problem on my Silver and found the
primer plunger was cracked. New ones are not expensive
and readily available. I need to pump mine about
7 to 10 times and then it starts on the first pull.

It has a new primer bulb and base, no string wrapped around blade, or shaft. Correct plug..plug looks lean. No debris under shroud..in fact a new shroud. Its not only somewhat hard to start initially, but after a while it just won't run, and it appears to be too hot.

Is your fuel tank venting properly?

I think the fuel tank is venting properly. I have several caps. I have tried removing the cap and replacing it, and even if the vent was clogged, that should have temporarily solved the issue, but it didn't. I have several caps, (several mowers) and I can always substitute one from a known working mower, but I believe the cap vent is open..

I have same mower and same problems about to sell it online now!

Since this never got resolved, I was just curious - when it stops and appears to be hot, does it still pull about the same when you're trying to start it or does it seem to be nearly stuck? If it still pulls freely, it's probably not severely overheated. The spark plug in that 'V' engine should be black, not look like it was in a 4-cycle. 'V' engines should be set-up to run 'rich' like the legendary Lawn-Boys.
I would look for an air leak in the intake tract, like at the carb base. It might show up more when the mower gets hot. With the mower running, Spray WD-40 on the suspected areas to see if you hit the air leak - the engine will change when you do.
Another prime suspect will be the crankshaft seals. These mowers were old even in 2010 when this thread started and they should have had these seals and possibly the lower bearing replaced already. A 2-cycle has to have a sealed crankcase to run. It gets pressurized when the piston goes down and it draws a vacuum when the piston goes up. If a seal leaks, the engine will tend to 'lean-out' or not run at all. Along those lines, the crankcase is in two parts with a sealant in the parting line. That could leak too. At least with the 'V', you don't have to worry about the head gasket - the head/cylinder is one-piece. But check the threads in the spark plug hole to make sure it's not sucking air there.
That's about all I have except for possible scored/burnt cylinder/piston.