The reason dealers can do this and you cannot comes down to one thing: they keep a claim on the car, and you would not.
When a dealer or a lender finances a vehicle, the loan is secured against it and that security is registered. If payments stop, there is a legal process for recovering the vehicle, and it works because the interest is recorded somewhere that a buyer, an insurer and the authorities can see.
As a private seller you have two options and both are bad:
Hand over the car and hold the title. The buyer has your car. You have a piece of paper. If they stop paying you are chasing a stranger for a debt, and repossessing is not something you may simply do yourself.
Keep the car until paid in full. The buyer is paying for something they cannot use. Almost nobody agrees to this, and it also raises the awkward question of who insures it and who is liable for it meanwhile.
There is no arrangement between two individuals that avoids one of those.