It doesn't seem to me that you're going to be able to solve this problem with potted plants. For suggestions that would work, we need to see the area, including some surroundings for context.

Yardvaark usually suggests taking several photos. Stand in the same spot, start at the left and take a photo. Move slightly to the right and take another. Keep on until you have photographed the entire area.

Do you own the home? Can you remove some concrete to plant a tree? Can you put up a arbor? Pergola? Canvas top gazebos are popular here, but you have to be able to tie them down. Shade sails are also nice. Can you attach lattice to the brick?

ETA: As it sounds now, nothing other than a cactus will survive the summer and it won't survive the winter. You still wouldn't be able to use the patio in the summer.

ETA2: Move the camera slightly to the right!

I agree. With the same exposure here, it's just sacrificing plants to the sun gods.

Why haven't you looked into one of those retractable awnings? There's a good DIY site I'm looking into myself. Maybe it's an option for you. Retractable canopies

Lots of plants, but I always forget to water for that important day or so and then they are toast, lol. Still can't sit out unless you shade all that concrete.

"Move slightly to the right and take another..." Actually, DON'T move. Stay at the same location and pivot the camera. If the camera changes location, the pictures cannot be put back together into a single, wide-view panorama. Mentioning the word, "panorama" gets people to thinking that their camera has a panorama feature, or they can make a panorama out of the pictures on their home computer. But this isn't a good idea because it ends up shrinking the individual pictures into a single, much smaller picture, which has the built-in problem of now we can't see details.

Agreed, there are plenty of heat tolerant plants for containers. But I got impression that the problem that needed solving here was protection from the heat for the patio users. I can't envision that solving that is easier by using plants in containers than it is by placing plants in the ground, since, ultimately, it's big plants that are necessary.

I used a couple of triangular shade sails as a temporary solution to mitigate some blazing summer sunlight. I have to say that they are overall inferior unless a person gets vastly oversized ones. They pull inward from the connecting points to such a degree that is is like trying to get shade from a skimpy bikini bottom. In the summer we can't sit on the patio long before having to adjust the chairs to a new location. I have plans to deal with this but it's expensive or involved, with lots of factors working against me. I'll be a while.

Here's some photos of the patio. The rear facade of the house is facing west.

One thing to note, the balcony is coming down this summer as this impedes too much into the patio below (it is about 5' from the concrete). Additionally, the strip of grass along the length of the patio will become garden bed this summer.

I see trees growing west of the patio. What kind are they? If they're capable of growing larger and producing shade, I'd say you've already implemented the long term solution, and now it's a matter of growing them on so they can do their job. A right dose of fertilizer at the beginning and middle of the summer and weekly, supplemental watering if it doesn't rain, always helps young trees to grow faster. Given the need for shade, I'd make the effort to see that the trees get love.

A vine growing on the wall of the house would help the house stay cooler in the late afternoon, but it's not going to help the patio. For that, you need either a vine or some kind of shade overhead, or a "wall" of shade along the west side of the patio. The latter would not shade during the day (when an umbrella works) but it would put the kibosh on the brutal afternoon sun entering from the side. You just need to rig up something temporary, until the trees start taking hold. That could be the shade sails which I earlier poo-pooed. Since your patio is rectangular, you could use the rectangular shade sails, which cover more area. They still pinch in though, so you'd need an oversize one. They need to be attached to stout, well anchored posts as otherwise the wind can wreak havoc with them. , Or, it could be an arbor on which a vine could grow. The latter has the benefit of adding some great looking greenery. There are many annual vines which can get up and running (produce shade) in no time. Probably none is faster than gourds. But there are others, many of them flowering, such as Heavenly Blue morning glory. (Be careful of others as some of them are WEEDY and difficult to get rid of.) Moonflower vine is great.

Great suggestions!
The trees that are there are Japanese maple, they have already been there for 5 years so I dont think they will get much bigger.
I love the idea of vines over an arbor or a lean to - my only worry is that the window on the bottom is our living room - I wouldn't want that to block too much light.
I do have moonflower seeds. if I planted them in pots, would they be able to climb the brick wall if I flanked the window with pots? Moonflower would get tall enough to stop the heat from reflecting off the brick at least?

Here is a view from the (soon to be removed) balcony. this is looking SW. as you can see the trees are pretty measly, and I dont expect them to get much bigger.

My other thought is to remove the mulberry tree (right in the corner of the fence), and replace it with some sort of columnar evergreen like arbour vitae. I know those can be tall and slim.... but that would be a long term solution as it will take years to become established (though that will likely be done within the next year or two)

The typical Japanese maple tree gets considerably larger than what you have. Unless these are some special dwarf variety, you should have hope. I stress, give them fertilizer, and supplemental water on a regular frequency, and you may see an entirely new growth rate. If you don't, I'd change them out for something that will work, since your need for shade seems pressing.

Moonflower vines do not grow on walls, but coil around cord. I can't imagine that if grown in a pot they will get as large or flourish. As a temporary solution, they'd serve you better grown in the ground at the edge of the patio. If you spaced out tall fence posts along the patio edge and soundly sunk them into the soil, you could connect their tops with a medium-light duty chain (like porch swing chain) spanning from post to post. From that chain, drop twine to the ground at 12" intervals, fasten it to short construction stakes pounded into the ground at 12" intervals, and you'd have a large, inexpensive arbor as wide as your fence posts are spaced. With either moonflower or heavenly blue morning glories growing on it, you'd have a quick shade screen, as these vines grow fast (with fertilizer and water.)

I don't think a single arborvitae off in the corner is going to help much with the shade issue. Other than that it would be OK.

May I suggest a 'southern' method to solve this problem. Vegetation in this situation will not work. Sorry, not an inexpensive solution. Construct a roof out over the patio. Then install three electric fans along inside ridge poles spaced evenly. If blackflies and mosquitoes are a problem you may decide to make this into a screened porch.

You need a pergola. It doesn't have to be attached to the house, but it does need to be fairly large.

"Vegetation in this situation will not work." I don't know why you'd say that. There are all kinds of choices between trees and vines that are easily capable of providing shade. It just depends on how elaborate or quickly OP wants to solve the problem.

Until you get it figured out, here's a temporary tip: Turn the sprinkler on it !

My 9ft living room window faces west and wouldn't you know it there is brick there AND a nice wide sidewalk there as well. Several years ago when watering flowers some flowers, it 'clicked' that I was cooling down the area. I turned sprinkler so that it would hit the house and relief was minutes away. LoL