It is difficult to give suggestions as we cannot see the space you have. However, I like the idea of a pop-up flat screen. This would mean that the TV doesn't become the feature in the room. Alternately - You could have a wall unit built where the TV is hidden behind doors. If you have the room, have matching cabinetry on either side of the fire-place.

WOOOOW!!! What a truly impressive home and interior design. Very beautifully put together! Thank you very much for sharing your project, it looks amazing. The Hamptons style is definitely a style that oozes a lot of understated chic and effortless sophistication. Well done!

Re purposing the space making it work harder. Lovely windows. Pot drawers work so much better. Fresh and bright. We replaced our pine doors to alabaster shaker style cabinetry with matching walls and ceiling. Rebuilt pantry with sliding door with sensor light and an island bench with drawers each side. Worth the investment of bespoke cabinetry.

You have a complex set of requirements that won't be adequately addressed in an online forum of opinions. In addition the photo of the style of kitchen that you like is in a much larger space than you have available, and the size of that kitchen contributes to its presence. Without borrowing some space from adjacent rooms for a wider, larger kitchen then I would consider a G-plan with the end of the G being bench height seating with half the stools with their backs to the view, and the other half with their backs to the kitchen, as in this photo. I would engage a design professional to visit, take your brief, measure up and come back to you with a few concepts for the layout. Think hard about how important the symmetry is to your kitchen as it can really hamper creative solutions that are outside the box. It is still possible to have a Hamptons style kitchen, that has presence and is of a formal nature that is asymmetrical. The rest of your home is not symmetrical, and forcing a symmetrical design into a pre-existing space only ends in compromises to the functionality. A good design professional (not necessarily a kitchen designer) will look outside the three walls that are constraining your thinking and options to create an elegant solution, but you will need to pay for this creative service. Best of luck, Dr Retro of Dr Retro House Calls/Dr Retro Virtual Visits.