That's a tricky set of conditions :-) A broadleaved evergreen tree large enough to shade a 2 story house are somewhat limited in number. SoCal possibly has one of the larger potential pools of candidates to pick from but still somewhat limited choices available. And most trees, even small ones, will develop a very wide spreading root system, well beyond the dripline or canopy spread. And all will take many years before they are of climbable size if that is even possible.

I'd look at Ficus rubiginosa or Magnolia grandiflora for starters.

I have been steered away from all ficus because of their reputation for root damage. Do you think otherwise? The magnolia is a possibility although its not my favorite tree. I wish I could plant an oak tree or a maple but I think the neighbors would kill me and I'd probably ruin the plumbing and foundation!

I just found a new website that let me search with a lot of different filters and it came up with 3 trees: Black olive (leaves appear to me much more green than the other ones I've seen!), Mexican avocado, and Japanese Evergreen oak. Any thoughts?

I tend to agree with you on the root issue! There's a massive pine tree in our backyard. Before we bought the house, we had the pipes scoped. Nada. Clean as a whistle and its been here for 30 years.

Keep in mind that a big tree is going to have a correspondingly big root system.

Look at Holly Oak, Quercus ilex. Selectree rates its root problem potential as "low". Very nice Mediterranean climate oak for So Cal. The native oak has a more assertive root system.

If you really have space, Podocarpus gracilior (now Afrocarpus falcatus) which is truly a magnificent tree, billowy and grand, like a green cumulus cloud, but it does get big.

Avocado produces heavy deep shade, and lots and lots of leaf litter which should be left on the ground to protect the delicate, fibrous root system from drying out. On the other hand, guacamole! Make sure drainage is very good.

If you are looking for a more comfortable home in the summer, focus first on shading the house's windows before the time it takes a tree to shade the house itself. In SoCal the windows are what let a significant amount of heat into a house, if the home has decent insulation.

Yes, any tree is going to take many years to grow large enough to produce a pool of shade big enough to shade a house. And if the climbing is supposed to be done by children that are alive today they will be too old to be interested, if not actually grown up and gone before a tree planted now is big enough to offer much scope for that.

Maybe instead it might be possible to build a tree house in the pine, if it has the size and branching for something like that.

Otherwise if you look at a copy of the Sunset Western Garden Book there is a Trees for Garden and Patio table in the Guide to Plant Selection section near the front. You can use their Sunset Climate Zone system to see which trees they think are suitable for your specific part of California, a State which has many climates (the USDA Hardiness Zone system only addresses average winter lows, and not summer conditions etc.).

What about a California bay tree? It doesn't have particularly large leaves, but they are a lovely shade of green. It grows fast, will get big enough to shade your house and can be pruned to not be TOO big/tall, is evergreen, great for climbing (lots of personal experience as a kid), smells amazing, and is native to the state. I've not known them to have invasive/shallow roots.

Another tree that ticks all the boxes except evergreen is a fruitless mulberry. Big leaves, good for climbing, grows fast but caps out at a reasunable size, lots of shade. Only problem is that it drops its leaves in the winter.