Skoda Yeti Infotainment System faulty touch screen

[Skoda4]

I've just bought a low mileage one private owner 2017 Yeti Outdoor 1.2 TSI SE Drive from a highly reputable used car dealership based in Cheshire, England. Immediately after buying the car which included taking out a 4 year extended RAC Warranty, I was still on the dealership forecourt and attempted to put my home post code into the Satnav. Unfortunately, some areas of the touchscreen were unresponsive. I immediately brought this to the attention of the salesperson who also attempted to get the touch screen to work but failed. I needed the car over the Christmas period but was only given sparse details by the salesperson as to what the dealership will do to remedy the problem. 

The Head Unit is the Amundsen11 Infotainment system I think. My RAC warranty only gives me £500 cover for in car media and I've been told that Skoda would charge around £2,000 for a complete like for like replacement which obviously would mean I would have to fork out £1500 of my own money.

The dealership's website states that it only sells high quality cars and in order to ensure this, their vehicle technicians carry out a comprehensive specifically tailored inspection of all aspects of the car to make sure there are no ' nasty surprises when the customer collects the car' 

Unfortunately, I did incur a nasty surprise with the touch screen.

My question is:

1.Would I be within my rights to require the dealership to carry out the repairs completely at their own expense rather than telling me to use my RAC warranty?

2. Would I be within my rights to insist on a completely new unit rather than just a touch screen replacement?

2. Did me taking the car away from the dealership on the 21st Dec absolve the dealership of any responsibility even though the fault was present but not identified before I bought the car?

I love the car and would be reluctant to do a Voluntary Rejection of it if they refuse (I took out a part loan when buying the car to ensure a tripartite contract was in place as back- up insurance should the car turn out to be a bad one). 

Any advice would be greatly appreciated.

Thanks

[Simon TDI Sport]

I would insist that the dealership replace the unit at there expense as they sold goods unfit for purpose. As you pointed out the fault to the salesman before taking the vehicle. Then they should rectify it.

[Skoda4]

Thanks for your advice Simon. I'm in the process of doing that. Cheers