"I'm looking for the best scenic route with 1 or 2 breaks for a total of about 6 hours duration. That doesn't leave a lot of time for site seeing, so we would need to choose only 1 major attraction along the way and view the rest from the car window. We would leave York after lunch and arrive in Edinburgh after rush hour, but no later than 7 pm while it's still daylight in early June."
I, myself, would take the train. You say you just don't have all that much time. View the rest of the scenery from the window of the train instead of the car. It's more restful. You are already doing plenty of driving on the first leg of your trip. I am a huge fan of trains, so I would ditch the car at York, provided you don't have so much luggage that you cannot manage to get it all onto the train.
To me, the most interesting sight between York and Edinburgh is Hadrian's Wall, but you would need much more time to see that than you say you have. If you choose to drive the car on to Edinburgh, if you take the coastal route, you would pass through South Shields (a coastal village), where you could stop at the Arbeia Roman Fort (reconstructed) and get a good idea of the history of Hadrian's Wall. Continuing north on the coastal route, you would come to Bamburgh Castle, my choice between the two mentioned. Bamburgh Castle is 70 miles from Edinburgh. It is overlooking the ocean and a pristine sand beach. It's difficult to choose a favorite castle in England, but for me, this might be it. Stay at the Mizen Head Hotel (we loved this hotel), walking distance to Bamburgh Castle.
"We would leave York after lunch and arrive in Edinburgh after rush hour, but no later than 7 pm while it's still daylight in early June." You are going to run out of daylight doing the scenic route by car. Especially if you do a stop at Bamburgh Castle long enough to do it justice. To spend the night here may be wise.
Again, the train is the swiftest way from York to Edinburgh. If you choose to go by train, you'll just have to return to do the other stuff! There's lots to do between York and Edinburgh; at least a week's worth! :)