I do not recommend removing the solenoid except for brief tests. There are reports on the internet of it being a troublesome component, and it often fails partly rather than completely, so it could emit a click on test but still be useless. You can check it though, by following two steps. First, replace the solenoid temporarily with a suitable screw or bolt and see if the engine runs. If it does, reinstall the solenoid, apply 12 V directly to it, and try to start the engine. If it starts, the problem is in the wiring upstream of the solenoid. If it doesn't, the solenoid has failed.
