The last two cars I have bought were new cars, and partially financed through the dealer. I did it because the rates offered me were far better than any bank or credit union could offer. One was for 0.9% for 3 years, and one was for 0.0% for two years. I was approved for the financing before I turned over my trade in (only on one of those purchases) and before I picked up the new car. I cannot imagine even wanting to drive off the lot without knowing that the financing was a done deal.
I did once have a deal almost fall through, years ago, after they suddenly balked at the last minute to honor the trade-in amount that they had offered for my current car. On the day I arrived to turn in that car, and pick up the new one, they said the condition of the car had changed (it had not) and they lowered their offer by $500. I said "forget it" and started to leave and suddenly the sales manager ran out and said "ok, ok, we'll go with the original offer".