What do you mean by bottom ventilation. I do have cross ventilation at the top.
Heat moves up (with ammonia and humidity). When air moves out of one area, other air must move in to replace it. In a closed box with only upper ventilation, some of the same air that's attempting to escape through the vents will get sucked back down in the box by the negative pressure created. Forming a loop effect where only some of the heat, ammonia, and humidity is vented.
A chimney effect is exactly what it sounds like - lower ventilation serves as the intake of fresh cool(er) air which can rapidly move into the void caused by the hot air moving up and out the exhaust vents. That keeps the air changing even when the wind is not blowing.
The "cross ventilation" you were aiming for only works when the wind is blowing straight through. It's better suited for cooler climates if used as a standalone.
It also needs much heftier proportions to work well. The smaller exterior upper vent is limiting your air movement to just that size - equal amounts of air must move through intake and exhaust, so it's constrained by the limiting factor.
@3KillerBs has great illustrations -
https://www.backyardchickens.com/articles/repecka-illustrates-coop-ventilation.77659/
I don't think it's your fault - the internet is rife with "coop designs" that prominently feature closed-up pine boxes.
I guess because it makes us humans feel the birds are safe? But that's what HW Cloth is for.
Even if these designs didn't make the birds miserable up north (they do), they would absolutely torment them here where the heat is so oppressive. And it's going to be hot into November, I always find October the worst month. We plan our garden to harvest tomatoes in December when it starts to cool off a bit.
The least thing I would do is replace that access door with a fully HW Cloth + frame version, not just a cutout. Make sure you use the cloth and not chicken wire like your upper run walls are. Just in case a determined racoon or whatever makes it through the upper run walls, I'd also add the HW Cloth to your upper vent inside the run.