I think if it were me I would do a few things- firstly also post this dilemma on the US version of the site (when you set up your account you probably put the wrong setting and the site thinks you are in the U.K.) I would also be thinking about how you entertain- it it were me I would want a big kitchen island with stools opposite so cooking can be sociable and they can also provide casual eating at the same height as a chefs table further down the island. (I don’t know if you use that term but I mean a Chefs table to be a tall informal dining table with tall comfortable chairs for leisurely and sociable dining.) Personally I would put the formal dining table where the pool table is- I think a rarely used table is easy to keep well presented and makes a great first time impression if it is near the entrance- it is also nicer for guests to walk though a dining room than a living room to your personal space. I think I might use the wine wall idea as a divide between the pool table/formal dining area- with glass on both sides it just adds some separation without making the space feel smaller. As far as the nook in the middle I think how you use it depends on how you socialise- you could have it set up for playing cards, smoking ‘gars, star gazing.... but if it were my house I would have a round pedestal table in the middle and use it like a bar cart for your favourite spirits- with an ice bucket and some sliced fruit ready guests could help them selves which also makes entertaining easier

Do you require two doors into the kitchen/living space? If possible I would look at the options of removing the door on the right and moving the other door to left to line up with the front door in the hallway. It looks like you could pottentially gain a fair amoutn of space if this is possible.

I have these ones, they are good quality and very comfortable even though they don't look it as they are not 'padded', lots of colour options - https://www.atlanticshopping.co.uk/bar-stools/drift-oak-and-white-bar-stool.html I know they are not swivel but I find swivel stools very annoying and I specifically avoided swivel ones for my 8 and 11 year old as I imagined they would be constantly swivelling around in them which would drive me nuts. They have a good height on the back too which I like and you can lean back comfortably. I've just come across these, which go with your armchairs if you're looking for a match... https://www.atlanticshopping.co.uk/bar-stools/drift-oak-and-blue-velvet-stool.html re: kids dragging stools with legs will be no different to them dragging dining chairs with legs :-), they're unlikely pick the chair up and place it down nicely when moving, I noticed that even adults don't do that :-).

This is a helpful podcast if you are thinking about what size of kitchen island you want or what to go for: https://shows.acast.com/love-your-kitchen/episodes/exploring-function-over-aesthetics-with-amnda-hughes