My "off the cuff", best in class, SWAG suspicions are that you have a flaky key switch, or the magneto kill wire has become open somewhere between the key switch and the magneto.
Of course your engine will keep running with the battery disconnected, the engine uses a magneto based ignition system.
Magneto based ignition systems do not require any external power from a battery or alternator, the magneto is entirely independent from the other electric circuits on the machine.
Now, I don't know about the harsh clunk or bad ping your machine is making now when running, but it might also be related to a flaky key switch in that if the switch has begun to disintegrate inside and shed some metal fragments, those metal fragments could conceivably be bridging across from a hot circuit over to the starter control circuit intermittently, and the noises you hear are the times when the starter solenoid is trying to activate the starter.
You could very carefully disconnect the small wire from the solenoid after starting the engine to see if the noise continues with that wire unplugged. If you do this, don't let the bare end of the wire touch a ground or you will blow a fuse.
So your mower guy here to do all you described?
Was he there BEFORE or AFTER all this began?
If it was AFTER, did he hear any of the noises? and if so did he have an opinion (on the noise)?

My mower guy was here before all this happened. Since he left, I have mowed my yard 3 or 4 times plus my neighbors. I think it has to do with me cranking to start it with me not sitting on the seat. Maybe I burned out the starter or something. I did notice when I pushed the mower home from my 5 - 10 minutes on my neighbors yard when it died - the starter was very hot. When I tried to re-start it - the top fly wheel would just turn in 1/2 spins - like 12 o'clock to 6 o'clock - but I could here the starter engage - just not enough juice or power to really turn it over. I drained all the oil out of it and its black and real runny.

***" I drained all the oil out of it and its black and real runny."***
You mean you drained it when the oil was still hot from using the machine? Or do you mean you drained the oil while cold and it was still "thin"?
Cold oil that is as thin as water undoubtedly has gasoline contaminating it. Smell what came out to see if it reeks of gasoline.
If this has been an ongoing condition that had not been discovered, your engine may be noisy now due to damage to the crankshaft, or to the piston and cylinder walls.
It sounds like you really have multiple issues causing several independent problems, not just one, all inclusive issue.
With the spark plug out of the engine, will the starter spin the engine? Can you turn the engine by hand with the spark plug out?

The engine was cold when I drained it (next day) - and yes it does smell like gas. When I pull plug the starter still turns the engines over. I can't turn this one over buy hand because it has no pull cord on it. By the way - it is a Ranch King Model 650G205-C150X - not a Lawn Chief. Thanks for all your info - its nice to know what it could be causing the problem

Geez Louise.
Let me ask a bit more about how the engine acts when it can't be turned off. Is this happening at idle speed when you attempt to cut the engine off?
Does the engine seem to be running normally when you are trying to cut it off? Or is it running really ragged and stumbling?
When I asked about turning it by hand with the plug out, I was implying that your hand would be placed on the flywheel, not a pull cord, sorry for not being clear.
Here's where I am going with this, now that it seems like chronic gasoline dilution of the crankcase oil might play a role.
The engine might have suffered some severe wear over time and due to this wear, the engine may have become somewhat fond of drinking oil (you did not say, but I am making an assumption). The oil consumption may have caused heavy carbon build up in the combustion chamber, and some of that carbon may be staying hot enough to cause the engine to "diesel" or run on, even in the absence of spark, after the engine has been run long enough to get hot.
The glowing carbon deposits might also explain the way the engine began running rough and finally bogged down and died as you described.
Try to pick out the questions and answer them to give a better overall profile of how it is behaving and when.
Have you ever removed the sheet metal shrouding from the engine to check for blockage to cooling fan ducts? Rodents love to build nests in there and that will make an engine overheat. Sometimes overheat will result in dieseling also.
I don't know what prompted you to use the word "weird' in your thread title but it certainly is apt.

Well, I bought the rider 6 or 7 years ago used for $400.00. So, when I got it - I guess I got used to some fnctions not working properly. One was the throttle lever sometime would just stay open (full rabbit) all the time. And when I bought it - the guy must have disconnected the safety seat so the only way to shut it off was pull the choke all the way out. Sometimes it would backfire on shutoff. When I would start it - I would notice some white or black smoke on initial startup. But not during use. It ran like a trooped except for the blades wearing out because of the sandy yard I have ( plus the sticks i tend to run over) ooops. I would change the oil once or twice a year. I guess I did have to add some oil to the unit every now and then.
I guess since my mower guy just got it ready after the winter with new oil, new fuel filter, and new blades and a good look over - I thought it was running like it always had. The only thing like i said prior - was when I cranked the heck out of it trying to start it while NOT sitting on the seat (that got fixed last year so now if you got off the rider it died) SEEMED to do something to the starting aspect of it. Now it started hard - I had to keep the key held all the way to the right as the starter tried to start the thing and just grind away until the motor finally took off. Now - the seat doesn't kill the motor of I get off the thing & the key doesn't shut it off either. I keep the unit in my garage all winter and although I didn't winterize it with stabil or run the gas out of it - every spring I would charge up the battery and it would fire up fine. Of course then my mower guy would be coming over and changing oil and checking it over. I sounds to me as it is running like something is rubbing inside the mower making it make the NOT SMOOTH idle sound. It is hard to describe. I think he may come over tomorow and since I have all the oil out of it - we will add new stuff and go from there. I appreciate your advice and knowledge you have given me so far. Anything else you think it may be so I could tell my mower man?? Are all the things you mentioned above worth spending to fix this or are some cheaper than others? Thanks Dave Wisconsin

With it being a Tecumseh, and 6 or 7 years gone by since you paid $400 for it, I would have to say.......fixing that engine is pointless, if it has sustained some damage from gasoline diluted oil.
Your new revelation about the sound of something rubbing is quite disturbing as that also indicates damage, possibly the piston has begun to gall and is scuffing hard against the equally galled cylinder walls.
Only by removing the head could you confirm what I am now suspecting.

Well - after draining the oil - which was black and runny and smelled like gas. I bought some Lucas oil additive and new HD30 oil. Filled up the crankcase, and turned the motor over Hoping I still had a lawn mower! NOPE - she ran but think I either burned a bearing or did something real bad internally to the motor. I think what happened just so others know: when my mower guy changed the fuel filter, the other fuel filter being so old and plugged - actually kept the gas from running into the float/needly valve in the carb when motor was off. Now, with the new filter - when I had shut it off - all the gas was running in carb and down into the oil. So, I got a few hours of use on my mower since he chage oil and filter but the end was coming. I think for all users of devices that have fuel shutoffs - they should use them when your motor is not being used. This can prvent this from happening and wrecking your motor. Because you will never know until its too late - Dave

We have been preaching that about the manual fuel shut off valves for years, and every so often someone stands up and gives a testimonial. Thank you.

But you can rest assured, a new filter did not allow more fuel to drown your engine with gas.
The carburetor has a problem.