No paint of any kind will provide any insulating value. All that paint can do as far as temperature control is concerned is to reflect sunlight
Not quite...
There's actually a lot going on when a surface is reflecting some and absorbing some UV energy (light,) converting the absorbed energy to heat, and radiating that out as infrared energy. I won't explain (because I call my uncle the retired chemistry teacher to sort me out on such things when I need help, and so that some engineer who took a class on it 30 years ago won't skool me on everything wrong with my post,) but black body radiation is is a thing, and why flat black is the best performing surface for frying pans, heat sinks and wood stoves. Nuff said...
For this discussion, a roof coating's thickness, composition and gloss level would very likely impact performance (the coating's ability reflect light (UV,) to radiate heat back off the surface (emissivity) and also to keep that heat from being transmitted into the roof skin on which its applied,} even with all choices of coatings being the same bright white.
Theoretically, Tropicool having a flat finish and higher emissivity than paint, combined with an order of magnitude thicker application of a material that could keep the surface heat away from the roof skin better, should easily outperform even a glossier, thinner, more UV reflective white paint film. ALL engineering hits compromises in the field, though. If Tropicool quickly loads up with dirt and its emissivity and UV reflectivity are degraded, its theoretical performance could indeed be overwhelmed by typical field conditions.
Its a hugely interesting subject, and worthy of a Master's Thesis like this one here:
https://apps.dtic.mil/dtic/tr/fulltext/u2/a598855.pdf
Here's a much more practical paper from a thermal imaging company on how to actually improve emissivity cheaply:
https://www.flir.com.au/discover/rd-science/use-low-cost-materials-to-increase-target-emissivity/
Fascinating, but so what? Its all about compromise on the road. Most of us would rather plumb the coach than go down a rabbit hole like this one, which is why taking the experience and advice of a fellow builder in reaching your own solution is a very efficient way to make decisions.
For myself, nearly the entire roof of my build will be covered with solar panels, (set 6" off the surface for a number of reasons) so my priorities are heat rejection, to be sure, but mostly to make certain that area stays clean under the panels and doesn't ever leak, which would be a severe PITA to correct. Haven't decided what to do yet...
Everyone does what works for them for tons of reasons; or it doesn't and then they redo it with a better compromise for them. We here have much more in common than what we so easily disagree about online. Try to remember that, everybody.
Over & Out...