You'd have to remove some pavers, but artificial turf is pretty popular right now for play areas.

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I wonder if you can install the turf right over the pavers? That was my first thought too. You also could get large planters and add small trees, tall bamboo, succulents and shrubs, etc. Small above ground pool or recirculating fountain? Depends on your budget, I guess. Cushy upholstered furniture looks inviting....Umbrellas or look at temporary pergolas/shade structures. I've seen some nice ones using large pots or bucket filled with concrete to hold upright poles to attach shade cloth and/or string lights over an area. This is a little elaborate, but depending on your budget, you could construct a wood platform and create a shade structure on that.

this could be more attractive, but just showing buckets of concrete and pvc shade structure

fast growing bamboo, but put it in large pots:

succulents are easy care, low water option for plants

large metal planters, but I'd worry about them heating up too much unless shaded

Removing some of the pavers, especially against the side of the house will allow for better drainage of your yard - a good guard against termites and stucco cracks. A pad of turf somewhere would be a nice pop of color and useful. Look into hopseed for along the wall. They are quick growing for privacy and easy maintenance. They also do not flower, so they do not create a mess.

I'd be wary of removing any hardscape in a rental. Or is this a home you own but rent out? I assumed you were the renters.

Youre right, were renting so im truing to figure out solutions without digging up the pavers, but we’ll be here for a few years so willing to invest a little to make it more livable for the kids.

I was thinking soft shade structures too. Planters if you can, with low-maintenance natives.

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I use a shade sail on my deck an it makes a tremendous difference in the heat factor. Mine is blue, and I think the color automatically makes you feel cooler.

Our neighbors installed what they called "high end" artificial turf in their front yard (San Diego area, not even the hottest part), and it melted. It was hot on the kids' feet. I wouldn't have it for dogs, because I have read you have to hose urine off of it or the yard smells of dog urine. And about that, we are not supposed to even use water like that in California, so there's that.

You need shade. Perhaps the Landlord will let you plant some fast growing cypress trees around the perimeter. They don't need too much water, and they will cast long shadows. Yes, those sail-shades are a great idea. Just having a few of those huge Costco umbrellas scattered around that can be tilted to follow the sun would help temporarily. Lastly, Costco sells these nice-looking gazebos that you might look at.

Local utility company where I live says turf is hotter than cement and discourages use.

For sure go with moveable things that you get to keep. Sun shades, umbrellas, outdoor rugs, cozy seating, potted trees and plants. You could make this quite lovely and it would all be yours if you move.

When my mother was a baby in the crib, my dear grandfather gave my blessed grandmother a wad of money and told her to go shopping and relax and have a lovely time. While she was gone, he had a crew come in and cement over her immense collection of hundreds and hundreds of irises, the entire backyard! Everyplace he would have had to mow. He said it was so that the children would have a place to play. We heard that story a Lot from Granny.
Good luck with your dilemma and for anyone else thinking about cementing overy your spouses favorite place? You will be remembered in loving infany.