For the corner, how about an Indian Hawthorne 'Rosalinda'? It gets about 12-14' high. Nice flowers, hardy to 10 degrees F. Gorgeous plant. Wish I could grow it here.

Some pittosporums can get to be 'tree-like' with training.

Yaupon holly 'tree' seems to be able to take the heat.

Carolina Cherry Laurel 'Bright and Tight', but it has berries that might stain your nice patio/walk.

Are bottlebrushes hardy there?

Fern Pine (Podocarpus gracilior)

Sweet Bay (Laurus nobilis)

Go to your local nursery and ask there. They'd have some good ideas for your particular climate.

Bronze Loquat in standard form looks like a possible choice.
I will just have to give it a lot of water in the summer since we have very hot and dry summers.

Here is a link that might be useful: Video

Will the people over the back be happy having branches on their side? Anything planted in that space will hang over the fence the other side.

I'm on a corner lot. Branches will not be hanging over anyone else's yard.

OK - I thought the rooves were on plots adjacent to your garden. I can see a gate there. Is there a footpath the other side of your fence? Don't want to poke anyone's eye out. (Speaking as someone who walks to work and has to dodge overhanging foliage. Worst when wet.)

There is a road between my house and those rooves you see in the photo. On the other side of the fence is a sidewalk and then a narrow street before it reaches the next person's property.

From a landscape designer's perspective, that location is simply too tight to accommodate any sort of tree or large shrub that will provide much in the way of shade. You are looking at something columnar.......and even that is pretty restricted by the space. Many columnar plants eventually get quite wide.

Have you considered bamboo? A) it's evergreen. B) you can select to the height you desire, from 6-8 feet to 25+, providing both immediate privacy screening as well as shade. And C) it's as narrow/columnar (or wide) as you allow it to be. Planting it in a container adequately confines running bamboo or clumping bamboo can be planted inground. There is more than enough space in that location to accommodate either choice.

I've looked at photos of them planted on strips of land smaller than the space I have. Supposedly the trunk and roots can grow in a 2x2 cut-out in concrete and I have more than 4X4 space.
I'm planning to get a 15 gallon pruned into a standard tree form for eatch end of the patio and use a patio umbrella for a couple years while I wait for it to grow out.

I would just have to make sure that the lowest branches start higher than the 6 ft fence.so there is space for the branches to spread. I'd want the lowest branches to be at least 7ft off the ground, not just to clear the fence, but so it is easy to walk under and it will need to be trimmed back if the branches hang into the road where there needs to be 14 feet of clearance,
So, to keep it even, low branches will need to be kept pruned on both sides of the fence up to 14 feet over the street. It should not ever grow over 25ft and I plan to keep it pruned so that branches don't stick out more than 10 feet diameter from the center of the plant

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I can see what you're getting at and it is true that there are trees which will fit in there - we have a lot of similar problems in our tiny gardens here. I have a Sorbus vilmorinii, a holly, a weeping birch and a medlar all in similar spots. They do provide some shade but we have very low sun much of the year. In fact I try to avoid creating shade in my garden! You say there is a side walk the other side of the fence and you are right to lift the canopy. But I think it needs to be higher than 7 feet. A 6 foot person could still get a branch in the face if it is hanging down from rain or weight of foliage.