What state is your location?

You might want to look at a coating formulated to keep surfaces cool. NOT PAINT.

I do not know if this will work for you, but your search for a material has to be more than just paint.

Additionally, consider installing something that is a barrier from the sun like a solar sail, umbrella or pergola.

I see no reason why your couldn't use a product made for DIY like Romabio limewash to achieve the effect. Experiment in a corner...really, what do you have to lose? AI agrees....

Sure you can use Romabio on the stone but there's no guarantee that the temperature of the stone will be less when it's exposed to full sun

have you considered doing a pergola cover so it's not so hot?

thousands of ways to do this. donesn't have to be super fancy. you can even buy DIY kits

or maybe a shade sail? during the cooler temps you could remove it.

i think they have some attatch right to your house and retract

Thank you! We will try the cool coat and experiment with rimbalo on spare stone. We have an existing enormous pergola with a wood floor the previous owner built so adding another won’t work, it’s a fire pit area as well so a sun shade is also a no go. I’m thinking selling the stones is the most ultimately economical thing to do so maybe we’ll just do lighter polymeric for the time being.

We’re in southeast North Carolina

I would not use Cool Coat on a patio. This is a high build acrylic elastomeric coating. Like liquid rubber. It is a wall coating...or sloped roof coating. Not to be used anywhere water will pool. 5 yr warranty.

There is a Cool Coat deck product but I would be very, very hesitant to use even that on flagstone with a polymeric sand fill. Because of all the edges on flagstones where I would expect peeling to start.

Rombaio FAQ: Romabio Masonry Paints are not recommended for use on floors, high traffic areas or horizontal surfaces out in the elements.

it’s a fire pit area as well so a sun shade is also a no go.

That leaves vertical shade strategies & transpirational cooling as about your only option. Trees/ non invasive bamboos, tall ornamental grass, trellis & climbers, perimeter vegetation. But vegetation is also going to increase humidity, block breezes and attract mosquitoes...likely all undesirable given your location.

Perhaps consider a weatherproof exterior fan?

Any approach that involves making the stone lighter in color is also going to increase the glare factor.

We have an existing enormous pergola...

Then use that as your high summer retreat and the flagstone patio when the weather is moderate?

Add a roof over it - no matter what is there it will be hot in the sun. Wear shoes too. Consider a sunroom or green house. . .

Does anyone have any thoughts on the mortaring/grout wash? Visually we would like to lighten it up regardless of heat. We were aiming for French provincial and it feels more 70’s…

That Leanne Ford Room is lovely, (I'd love that in my own home...sigh...) and I don't have the expertise to weigh in on the grout/paint/color-lightening issue... but what if you try an entirely different direction since your patio is out in the hot sun? Perhaps planting some sort of low growing ground cover between the stones and utilizing some smaller trees in large planters could bring the temperature down enough to invite more welcoming use for you? I'm sure a local nursery or even Ag. Extension office could advise you on what types of plantings might work for you. You might also consider adding a small recirculating solar powered water feature since this could provide some degree of evaporative cooling effect. I would personally love to have enough stone here in my East Texas woods to construct some patios and walkways. Good luck, and enjoy your nice patio!

https://www.houzz.com/discussions/6114714/mortar-wash-brick-wall-and-ground

try reading this one^

I'm seeing that limewash won't hold up to foot traffic. But, you may want to try it out and see. any spare pavers ? Romabio has a small sample jar you could try

https://www.houzz.com/discussions/5918688/can-i-whitewash-a-concrete-paver-patio

Thin down any white porch paint and rub it in and rub some off. It's paint not rocket science people. Hasn't anyone spilled white paint and tried to clean it up with rags? That's the method.