Flagstone patios are wonderful. The only downfall is that the stones have uneven surfaces which can make for difficulty leveling tables and seating.

Personally, I do not like concrete. No matter how carefully it is installed, ground settling, climate change etc. cause cracks to develop. Consequently, I installed pavers. They can be lifted and re-leveled is necessary to maintain an even surface. I believe there are even flagstone looking pavers made now.

The base compacting below the install is the critical element and if you have experienced contractors to consider, ask them what they believe works best in your climate.

Good luck.

When I think of mid-century modern, I think of natural materials in a modern design. So that points to flagstone. But I'd do square and rectangular flagstone, in some sort of tile pattern like pinwheel or versailles, not the random size/shape stuff that's often used.

Why not stamped concrete? You could do flagstone or any number of other textures and patterns.

Another vote for pavers (especially of you have clay soil or trees with shallow roots)

I don't trust concrete. Seen it crack way too many times even on newly poured patios or driveways. Go with pavers or flagstone, especially if, like salmon said, you have any tree roots nearby or clay soil.

I second pavers as well, for all the reasons above. They look great, low maintenance, little cracking (though you might get settling, easy to fix). Might be a bit more expensive than the other options, but it pays off after a lot of years.

Here is a link that might be useful: Concrete Pavers Guide