It might be your starter relay is bad. Or solenoid, Same thing.
I would reinstall the starter, leave the cable coming up from the starter loose from the relay. Jump from battery positive to the cable from the starter.
Another thing, the starter grounds to the engine, the engine grounds to the frame, frame to battery. On the rear engine mount, the one that is two kinda triangular pieces and three bolts. Remove it, clean the contact points on the two mount pieces where they touch the frame. Clean the places on the frame the mount pieces touch. Clean bare metal to clean bare metal. This will help correct a weak ground. I would also do the top mount. All four pieces and the bolts that hold the pieces together.
Front mount too.
A thin coat of grease on all these contact points will help prevent corrosion that will stop the ground.
Oh one more ground point, when you reinstall the starter clean off the contact points where the starter touches the engine. Clean bare metal to clean bare metal. Where the battery ground strap hooks to the frame, another place for clean bare metal to clean bare metal.
With the starter off, clean the cable ends, the stud the cable hooks too.
All electricity travels in a circle. All your power side can be good but without good grounds stuff don't work well.
If the starter spins the engine buy doing the jump thing then your starter relay is probably bad.
Leo