Any thoughts on capital pear tree for zone 8-9 in California?

Have you considered a palm? I'm in NC, but that fence/patio situation sounds perfect for a Sabal palmetto. They are evergreen, relatively clean and low maintenace, here you could get them put in at a size with the crowns over the fence height for a reasonable price (around $200 ea). Palms in general don't need a huge area to root, and once they grow upwards, they can never ever get any wider at the base. (perfect for having limited space down low). I don't have a clue about growing Sabals in CA, or the price of a 10ft one there. There are a handful of palms that could work for you in z9. I know you were asking about shade trees, but if you don't hate the idea of palms some could be well suited for your space. Just my palm biased thoughts.

Palms would be out of place in this area and not provide enough shade or patio privacy.
I am in an area that is 8/9 borderline.
I found a site that says the capital pear tree is supposed to be 6 feet from fences and I only have around 4x4 ground space to plant in so there is no way it can be 6 feet away from the fence. So, that one probably will not work either.
I'm looking for sun protection during spring/summer/fall and added privacy.
So maybe some kind of hedge could work instead of a tree, except hedges generally do not grow in a way that provides shade like a tree would.

not sure about your state / area but the local botanical garden lists this invasive as a NO PLANT for Missouri. Besides that, until proven otherwise, I bet this Capitol Pear will hold a split apart and attack nearby fences like its Bradford relative.

Is this a 4x4 square of dirt Surrounded by pavement you will be planting a tree in? There are columnar oaks and maples but I would tend to want to stay one step smaller if planting in such a tight spot. Acer gresium, paperbark maple, or the right Acer japonicum, Japanes maple cultivar might look great if hardy for your summers.

Here is a link that might be useful: MOBOT capitol pear

I found out that the capital pear tree is probably going to be too large to fit in that space.

It would replace that small plant and moved closer to the center of the corner..
There is fencing around it on two sides and hardscape on other sides. So, only about 4x4 feet is usable.

What about a Cupressus sempervirens?

Cupressus sempervirens looks like it may fit in the space, but it would be so narrow at the top that I don't think it would provide shade.
I'm looking for something that doesn't need a lot of ground space, but has a canopy overhead so it can provide shade below it.

That's a tough one for sure....

Maybe bronze loquat?

Here is a link that might be useful: Loquat Video

I think your parameters are mutually exclusives. The 4X4 space is not large enough to plant any tree that is going to provide any shade.

Have you thought of large bushes around some trellis/arbor type structure? The bushes would give you the green and color and the structure would give you the shade.

Is it possible to remove the fence to open up the area and provide more room for a shade tree?

I went to a nursery and they said they many of their smaller trees that have a maximum height between 15 and 25 feet would fit in that space.

The amount of shade would be limited, but I I don't need to cover the entire patio in shade,

I can't remove the fence and I don't want to install anything like a trellis.

Possibly standard full-size Crepe Myrtle bush. Tx. Redbud - train to single leader. Desert Willow - Lucretia Hamilton cultivar.
Mexican Plum. Flowering Peach tree.