First of all
Ok. The battery going dead could have played a role in the machine dying. It sounds pretty obvious that a new battery helped. Here are some things to keep in mind
1.) The ignition system is a magneto, so it requires NO VOLTAGE to run. The key grounds it to kill the motor. Along with ANY other safety device. (I.E. brake switch, clutch switch, seat switch.....)
2.) This motor has a fuel shut-off solenoid, called the "afterfire solenoid" in the electrical schematic. This DOES require voltage to work. IF, the battery went dead, and it dropped that solenoid, it would die. ALSO, if that solenoid went bad, it will die and will not restart.
3.) There are like a million safety switches that can kill the motor.
Here is what I suggest:
1.) Disconnect the "kill" wire from the magneto and see if it will start.
If it does: Find the problem in that circuit.
If it does not: Hook it back up, and go to #2
2.) Check to see if the fuel solenoid at the carb is getting power.
If it is: Check to see that the solenoid is working. If you hold your hand on it and turn on the key, you should feel it "click".
2A.) Another way, is to remove it carefully (It will have a spring) take the plunger out, and reinstall it. Then try to start it again. If it runs, the solenoid is bad. Replace it.
If it is not getting power: Test that circuit and find/repair the failure.
Cranking issue:
1.) The motor has a compression release, it may be bad.
2.) Check/ clean all power and ground connections. Make sure specifically, that the ones at the battery, starter solenoid, and starter are good.
3.) It is possible the starter is weak, or you didn't buy a large enough battery. They have a horsepower rating on them usually.
If you try all this, and still have no luck, or have questions, come back and we'll try to help!