There is no solenoid on a stomp starter. None. You might have some bad cables. They will corrode up under the insulation. You might have a bad switch on the starter. And the brushes might be hung up or worn. You might, just might have bushings worn so bad the armature can touch the case magnets. And of course the battery might die under a load. All can be fixed and all can be tested. The "click" you hear can be a bad connection, a stuck armature or a bad bendix. However, the stuck armature will cause some damage. And it can be expensive to fix. Pull the starter. Two bolts and maybe, two cables, wire. get it on the bench and look at it. Some have been filled with water from just sitting around. If it will turn and if it will motor on the bench, like you have posted, then look for wear on the switch and then the cables and battery.

Steve H