My understanding is the surface area, water flow, air flow for gas exchange, and lighting is best on the waterfall style
Surface area is simply what size you make it. Does not matter what style.
Water flow only needs to be "somewhat", to circulate nutrients. But water flow does not remove the boundary layer on the algae; this instead requires air/water turbulence interface.
Air flow is not so much for gas exchange, as it is to create air/water turbulent interface to remove the boundary layer on the algae. And for this, bubbles are best, followed by waterfall, followed by horizontal river (no dump bucket). If a dump bucket is used on a horizontal river, it becomes as effective as a bubble upflow (because of course, there are now bubbles).
Lighting is actually worse on a waterfall, because the growth mats down and blocks the growth. Blocks water flow too. This is why waterfalls get brown dead spots in the middle of the screen. A super strong focused light can help solve this, by over-lighting a middle part of the screen so growth does not grow there. Bubble upflows are always under water, so the water supports the growth and prevents matting down.
why not just put the roughed up screen material mounted horizontally
Yes it will work if the light is down there too. The challenge though is bubble spreading, which will leave grown areas as upside down islands with no more bubbles, just light regular islands on horizontal rivers. Better might be to just attach strings the eggcrate and let the bubbles and growth go upwards.
but with a normal powerhead aimed at the surface of the water creating waves.
If the screen is at the surface, this will work and will be someone of a cross between a horizontal flowing river and a dump bucket. Waves must be strong and reach the screen though.
My experience is that chaeto thrives even in a 4" thick mat.
Depends of course what you mean by thrive. The inside 3" is certainly lacking; it may not be brown yet, but it's very shaded. Shade does not generate much photosynthesis. It can probably hold it's shape, however, which is why you can pull it out in one piece. Just remember you need 5X the chato mass to equal biomass growth rates of rapid GHA growth.
Maybe the green from the warm white is what penetrates deeper
It might penetrate, but chlorophyll does not use green at all. Only red and blue. White is just for watching (and white LED's don't really have a full spectrum anyway; just a "trick" spectrum to fool your eyes. Doesn't fool algae though). Almost every time someone had chaeto growth problems with white light, after they switched to red, the problem stopped.