Friction does not hold your car to the surface of the road. The ability of the tyre to key into the surface is grip.

The shortest way isn't the fastest way. Doing what is essentially a U turn by hugging the inside is the slowest way possible. The fastest way through a corner is the straightest way with minimal turning. The straightest way is to go wide, then hit the apex and exit wide. This is something you've already even demonstrated to him. I'd just not bother if he is still going to argue it.