Its a Toro 708cc model 127-9040 with a 2 barrel carb.
On a twin barrel cab, one side not running can be because the carb is bad on that side
Things like the main jet not installed or blocked on one side only.
With a runs on one side only problem the trick with diagnosis is to start by separating fuel from ignition
Start with stuff you can see, like valves moving
Slip the rocker covers off & plugs out then rotate the engine by hand at least 2 full revolutions while watching each side
The rocker movement should be identical, inlet to exhaust and left to right
B0th vales should move the same distance and sit at the same position when slack.
Next try without the kill wire on the magnetos ( check it will stop from the fuel solenoid )
NExt swap the coils over , note they have to go in the same way up
Problem shifts side = bad coil
No difference then the problem is in the carb
