How do you feel about a backsplash? And some undercabinet lights?

Do you mean brown paint is all over? Or wood or furnishings or what? Just doing your kitchen walls gray will look a bit off imho, especially if you have the kitchen open to another room.

I would do a backsplash to match your cabinets.

the whole house is conne ted to the kitchen, so initially the whole house would be a grey

I get that you're tired of the brown tones. I would look for more of a neutral and not a cool gray. You have to be careful as because your floors are a beige color(?)...

This is Agreeable Gray (Sherwin Williams)...

I will say, though, that suggesting paint color is not something I ever can do. It's dependent on existing light at various times of the day. Plus what I see on my monitor isn't the same as what you see in real life.

Your best bet is to try various neutrals in your own space...

Here's some good reading (her whole blog is good and easy to understand...)

https://www.kylieminteriors.ca/the-4-best-warm-gray-paint-colours-sherwin-williams/

Can I suggest (since you're embarking on this fabulous transformation), removing the wood valances from the window and the built-in in the eating area? If you like them then by all means keep, but they are a particular look from a particular time.

Also, have a look at these. It's an idea to help give your built in some punch and is one of the easiest high-impact diy's...

These colors are vivid, but I wanted to emphasize the impact. Of course you can do any color you want...and it's a future project to think on... :)