It's not his grip.
In the second video, when you have him swing the ball, notice that it starts inside (in the front) and swings outside (to the rear) - NOT straight as it should.
Now - notice that his swing IS ON that same path.
Instead - have him hold the ball farther to the right, then, as he drops the ball into his swing, it will actually be in the line you want.
BUT**** the main problem is not that either. The main problem is his steps. Nothing starts at the right time and he is trying to muscle the ball through to the end.
He's using a four step approach - so we'll count from there.
First step - push away. Ball and foot should extend the same amount AT THE SAME TIME. His foot is planted before he really starts moving the ball.
Second step - drop the ball into the swing. As Norm Duke says, the ball weight should be on the right arm when the second step hits. Your guy has a HUGE hesitation between first and second step (likely to fix the fact that his first step is so early) so that the ball is at the bottom of the swing when he starts his second step. So now his ball is WAAAY ahead of his feet.
Third step - this should be a stutter step, but not quick. It sets up the slide. But he's running because his ball was way ahead of his feet and he feels like he has to catch up. At this point he should already have the ball at the top of his backswing and be ready to start the delivery, but he's now ahead of his ball.
Fourth step - this should actually be just a touch ahead of his ball, but since he's been running, he's waaay ahead of his ball. Now he has to grab his ball and yank it through, pulling his ball to the left.
Don't give up on him. Give a shot at watching some Norm Duke, starting with the setup, through the stance and pushaway, etc. https://www.youtube.com/watch?v=QxmjBuun2zc