I like the options you provided. I think they coordinate the best with your home's traditional style. The thing throwing off the rendering is the pergola's proportions. I have a feeling the columns won't appear as large in real life as they do in the rendering...so, keep that in mind. I prefer the look of the flatter roof but if you are in a region with snow I would make sure to build with a pitch.

We're in California, so we're considering a flatter solid roof or more of an open traditional pergola.

I'd go with the pitched roof just because I think it looks better. In any case, have a solid roof, in which case it isn't a pergola, but a pavilion. Pergolas are open on the top and usually have some sort of vine-y plant material growing on the roof area. This can allow both sun and insects to be overhead, which I don't find particularly comfortable. A pavilion has a solid roof but open sides which would work much better with the California sun!

I agree with Diana Bier - a pavilion instead of a pergola since a pavilion will provide true shade. Since you like an outdoor kitchen and dining area why not install an one at the rear of your pavilion, say along the farther wall (as seen in the photos). You could enlarge to pavilion to include dining area as well as other seating or you could have dual purpose seating/dining areas.

Thank you both, I tend to agree. I'm a bit concerned with the 'size' of a pavilion in relation to the space -- do you think it would be too visually heavy given the relatively small space of the yard and proximity to the house? Especially with a pitched roof, which makes it even larger/taller. That's my primary concern, just the visual weight of such a structure in the space. Alternatively I'm considering no structure and just a series of 3 large umbrellas and planting some shade trees (obviously it would take a bit longer to get to the desired state of shade with trees)

My current thinking is to build a covered pavilion the width of the pool, but only about 12' deep, and creating an uncovered kitchen/bar area behind the pavilion away from the pool.

A gazebo?

The value of having a structure to me far outweighs any heaviness, which I don’t think it will feel, but only enhance your backyard and create interest.

Suggest an open beamed pergola stained vs painted so it feels natural and blends well into landscape.

If there is not enough shade with beams alone, down the road you could add a retractable canopy (manual or automatic) and at night it’s nice to pull back to see the stars. Less expensive is adding draped fabric

I would also build as large as possible so you have more options in regards to set up and usage.