The next step would be to make sure that you have 12 volts to the solenoid. Red to grey wire, and black to black wire from your meter. if no voltage then move the black lead to a known good ground. If no voltage in those test then time to start backtracking the wires. the 6 pin connector between the engine and the mower wiring harness is a possible place for problems.
If you have voltage then remove the solenoid and make sure the plunger can be moved by hand freely. If it moves freely and you have 12 volts then you can assume the solenoid is bad. You can substitute a carb bowl screw from another carb that didn't have solenoid, or if in your case the plunger doesn't work with battery voltage applied then you can clip the end of the plunger off and then test to see if it will start. worse case scenario is a loud after fire from the muffler when you shut if off.
If you have voltage then remove the solenoid and make sure the plunger can be moved by hand freely. If it moves freely and you have 12 volts then you can assume the solenoid is bad. You can substitute a carb bowl screw from another carb that didn't have solenoid, or if in your case the plunger doesn't work with battery voltage applied then you can clip the end of the plunger off and then test to see if it will start. worse case scenario is a loud after fire from the muffler when you shut if off.