I would guess someone had someone run the numbers and decide that someone buying a car on a credit card is higher risk overall.
I have a secured car loan like most. The interest rate is really low, like 2.9%. If someone were to HAVE to use a credit card to buy a car, odds are they are higher risk. The interest will be somewhere around 20%, give or take a few points. We are getting up to the rates of “buy here, pay here” places at those numbers!
Then there are the people who know how to work the cards for points, etc. That might be another reason they don’t like it because someone could end up with thousands of dollars of airline or hotel points for free if they work it right.
I just looked and it looks like it depends on the bank and the dealer on how much you can put on credit cards when buying a car. Someone looked at buying a $45,000 car on Amex, but the dealer was going to charge them a fee to cover the transaction costs of using the card. It ended up not being worth it as the travel points were not enough to offset the fee. I guess there is no set rule on this.
I bought a car over the phone not long ago. They let me reserve the car using a credit card but only charged a few hundred dollars. The rest was done using standard financing.