Broken brackets or bolts out of the torque brackets that keep the transaxle from rocking can cause a belt to come off the rear pulley. Lock the brake and see if you can rock the transaxle. Know that oem belts need to be used on these as well. Any work been done on this?

You have a gravity feed fuel tank AND the needle valve in the carburetor is not shutting off the flow of fuel AND the fuel tank emptied into the engine crankcase AND when you started the engine, the gasoline diluted oil foamed AND got sucked out of the crankcase to be burned in the engine. I wonder if the carburetor on this engine was transferred from the other engine that "ran out of oil"? If so, the score is now Carburetor 2, Engines 0. Will the engine turn or is it locked up? You might get lucky because sometimes the old "gas in the crankcase" event leads to such a fuel rich mixture going into the intake as the engine heats up and the gas begins to boil off, that it smothers the engine causing it to quit running. You need to hope that's what happened, or that the fuel tank just ran dry before the engine could lock up. Anyhow, determine if the engine will still turn. If it will, do the following in this order. (1) Change the engine oil (and filter if it has one). (2) Take the carburetor off and remove the bowl, float, and needle valve for a good cleaning and inspection. Clean out the entire carb using spray carb cleaner and poke a small stiff wire into all holes and passageways, followed by more carb cleaner. Compressed air is good to blow out all the passages if you have it. (3) Purchase an inline fuel shutoff valve, mount it between the fuel tank and the fuel filter. (4) Replace the fuel filter and fuel hoses. (5) Remember to always turn off the fuel shutoff valve whenever you cut the engine off. Get into the habit of doing that. If the engine is locked up..........perform the above steps 2 through 5 to your new engine before you run it

It would be better if you get the switch mount welded back in place and replaced that inhibitor switch. Sure you are the only one who drives it and are habituated to always pushing the clutch, but someday after all this has been forgotten, you will leave it in gear and hit the key to start it while you are standing beside the machine and away we go. What about comes the time you get ready to sell or trade? Better draw up some papers admitting that the starter inhibitor switch has been by-passed and get the new owner to sign them agreeing that he/she will never try to hold you liable for injury or damage. If it was as simple as getting used to always pushing the clutch when starting, the mandate for starter inhibitor switches wouldn't have been necessary. Because people DO FORGET is why memory and habit are not acceptable substitutes for genuine safety devices. Sorry if you don't see it for what it is.

I don't know what model briggs wheather it's OHV or L head?? not going to take time to look it up. Makes no difference in troubleshooting anyway. IMO I would first remove the drive belt to ilimate any outside source. If it still locks up or won't unlock I would then remove the valve covers and check the valves if it's an overhead valve model. If it a flat head or L head design I would remove the head and check out the combustion chamber. you can also drain the oil and filter and look for any metal shavings or metal grit a tale tale sign of MAJOR problems. Level low or little oil in the engine would a another sign of major problems. Black thick oil another sign of what caused the major problems from lack of pre-ventiive maintenance. Low chance that it was just metal fatigue or machining problem beings you didn't give any history of the LT. Still can't find nothing on the overhead valve model I would then remove the heads and look at the piston to bore, next would be removing the engine and spliting the crank case and looking at the connecting rods. You can get a replacement engine pretty cheap (depending on the model) either from SEW or ebay. probably less than $500 bucks if you handy with a wrench and understand basic wiring if you feel the LT is worth spending $400 to $700 bucks on and your time.