Hi there! I was in the exact same situation recently — looking for a gazebo that could really work for outdoor dining, not just lounging.
After a lot of searching, I ended up getting a 14' x 10' Hard Top Lean-to Gazebo, and it’s honestly been perfect for us.

We mainly use our patio for lunches, dinners, and small gatherings too, so I needed something that would give a lot of shade during the day.
The hard top makes a huge difference — even at noon, it stays super cool underneath. There's plenty of room for a full dining table and chairs without feeling cramped, and the lean-to style keeps the space open and clean-looking.

It also looks really nice next to the house and blends in well — not bulky at all.
I'll add a photo of ours set up (second picture) so you can see — hope this helps you with your planning!

Personally I would have a larger pergola that also covers the seating area to the left. That leaves the lounge chairs on the right in the sun.

A pergola will not provide the shade you need in that open space. Take a look at having a pavilion style top for full coverage from the sun. Make it wider to house both dining and lounging.

We don't know where you are, but if you live somewhere it rains, pergolas drip, and drip, and drip.

I agree something with a solid top. An outdoor kitchen is IMO best as close as possible to the kitchen but I have no idea where that is from the pictures . It is odd to see the loungers outside the pool area but hard to really understand the whole layout.

If you want shade (and it looks like this area of your yard is wide open in the sun), a pergola isn’t going to work for you. Unless you add some type of fabric to it. Though, of course, when the sun isn’t directly overhead this won’t work well either. (You didn’t tell us if this is the west side of your house or …..)

With you home's colors I would only consider a WHITE PERGOLA KIT.

This one allows you to adjust the ceiling vanes for shade

Has pull down screens to control the sun and the bugs.

I think your sketch doesn't illustrate a large enough pergola for the table and chairs.