It’s not about shade alone. A cliff or single tree provides shade, but a forest provides evaporative cooling during the heat across a huge area alongside shade, it ends up a noticeably different climate.

There’s some other effects such as photosynthesis converting sunlight into chemical energy which in the short term is like reflecting that energy into the sky. At night plant metabolism warms the environment slightly and blocking the sky reduces radiative cooling to space, but that’s generally a good tradeoff for comfort.

Counterpoint: Shaded spots at work parking lots in Texas fill up the fastest. Conspicuously so. Also, use of windshield visors is much more prolific than in cooler climates.

I can't believe your Texan friend never noticed those phenomenon.

I don't respect your friend's observational skills. But to be pedantic, shades are cool because the sun does not heat up the air, but heats up the ground beneath you and it heats up the air. The water evaporated helps in cooling us down.