*Did you check the oil level? Many of the engines have a low oil shut off. 4 OZ of low oil will do it. The symptoms are:
The engine may fire once or twice and not fire again.( the white puff of smoke) As you pull the rope starter the piston rod is splashing the oil,if it is right at the "low" point it will not fire as the splashing drops the oil level even more BUT if ohm'ed out at rest it will check out as conducting as normal, the oil level just makes the switch. The splashed oil has run back down into the crank case.
I have had this happen several times over the years. It drove me nuts until I added oil until it overflowed the fill port. The engines started instantly if they were not flooded with gas.
When you say all wires have continuity,you better not have continuity on the kill switch to ground or the engines run/stop switch to ground.