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I think what you are missing here is just how large a distance 1/6th of its orbit is. Also lagrange points dont pull anything. its a where gravity and orbital speed sync up. so shotting something at a lagrange won't cause it to orbit. you have to get the right orbit going with it at that point and then it will stay in that relative point somewhat. even at lagrange points some adjustments will have to be made. its an island of stability not an attractive force itself. any matter that hits that point and is not the exact right velocity won't stay there and even if it did it would lose it eventually do to outside forces unless it had some mechanism to stabilize.
My point is, material from the more distant regions of the protoplanetary disk that eventually coalesce into the planet have to move through the Lagrange points to get to the protoplanet. But once they’re near those points, the gravity of the protoplanet is unable to continue drawing them. So how does the protoplanet’s gravity attract material from farther away than its Lagrange points?
again its an island of relative stability. they would have to get to the point at just the right direction and velocity to be stable and even then radiaiton from the sun or other effects can push it out. its stable for a man made sattelite because we make sure the momentum and dirction are just right and make minor adjustments over time to stay there. an object in motion stays in motion and an object at rest stays at rest unless acted on by an external force.