Re-registering/unregistering a car after the death of the car holder?

Hello dear community, my father passed away a few weeks ago and we want to sell his Tiguan now. Since we don’t really know all this stuff, I wanted to ask for some advice on what to do or what to do most usefully. I will contact the car insurance company tomorrow and have the insurance transferred to my mother, which is the heir. Now to my question: How is it most useful with the registration/re-registration at the car S should the Tiguan be rewritten to my mother before it is sold? My thoughts are then regarding the resale, since then still an owner is in the papers and the car is then worth less money?! Completely unsubscribe is also somehow not quite useful, because if the car interested wants to make a test drive, is this not possible?! Thanks for their help… Maybe my thought is also completely wrong, or I have something not Thoughtful Lg