This is not the reason, it's to do with the fees and has been standard practice for ever.
Most garages will accept a deposit on a credit card and, providing that the car is under £30k, gives the buyer the same protection whether they pay the entire cost, or just a deposit, which can be any amount, even just a penny.
OP you can get the car spend onto a credit card, but it might need a workaround, eg put all your other spending on the credit card and use the money you've not spent to pay for the car. If you pay for all your groceries, fuel and general spending, you'll probably rack up at least £1k pm, then use the money not spent on these items towards the car.
Or you could ask your provider if they have a money transfer offer and use that instead, but there will be a fee for that, so you could look at a personal loan instead, especially if it's likely you'll be able to pay it off over a few months and can get a low rate like 3 or 4% APR.
But I think that second hand car sales are still booming so it's not like they're desperate. I've been tracking the price of cars similar to the one I bought a year ago and the same year/model I paid £8k for a year ago is now £9-10k from the same place, despite being a year older.