https://www.youtube.com/watch?v=Us2Z-WC9rao
Yes, a collision-only slingshot needs precision. But we're good at precision maneuvers with modern computers.
(it is correct that getting past Earth's orbit in the first place requires a bit of delta-v though, which I suppose I glossed over.)
That precision doesn’t get around the physics. Why do you think none of our missions to the inner Solar System have tried this?
https://www.youtube.com/watch?v=Us2Z-WC9rao