In the videos, I hear the tractor crank fine for a second, then slow down. Then you say it won't re start after you shut it off, won't turn over, like it's locked up. That sounds like the pump is trying to pump while it's cranking, it's not supposed to pump while it's cranking, the starter motor doesn't have enough power to do that.
The solenoid on the hydraulic pump is supposed to shut the pump off, it's called destroking because it's a piston pump and the pistons don't move when it's destroked or shut off. When the pistons don't move, there's no load on the pump and no load on the engine, and the engine cranks faster and starts easier.
I would have somebody crank the engine while you hold a piece of steel near the end of the solenoid, you should be able to feel the magnetism while it's cranking, and not while it's not cranking. If you don't, then check for voltage while cranking, and check the coil for resistance/ohms with the wires disconnected. Even if you do feel the magnetism, then the inside of the valve could possibly be sticking, you'll need some better pictures than I have to mess with that. But we can find it when you get to that point.
The solenoid on the hydraulic pump is supposed to shut the pump off, it's called destroking because it's a piston pump and the pistons don't move when it's destroked or shut off. When the pistons don't move, there's no load on the pump and no load on the engine, and the engine cranks faster and starts easier.
I would have somebody crank the engine while you hold a piece of steel near the end of the solenoid, you should be able to feel the magnetism while it's cranking, and not while it's not cranking. If you don't, then check for voltage while cranking, and check the coil for resistance/ohms with the wires disconnected. Even if you do feel the magnetism, then the inside of the valve could possibly be sticking, you'll need some better pictures than I have to mess with that. But we can find it when you get to that point.