Definitely getting off the original topic. But when you say, "as little thickness as you can get away with", well, I'm somewhat confused. You say that the heat dissipation drops off rapidly. Still, it doesn't go to zero, thicker is still better than thinner, yes?
In terms of thermal mass, yes. In terms of cooling, no; in fact, I have an anecdote for this exact question.
Neighbor down the street had a supercharged Mustang he would race down at the local airstrip when those half mile races were going on. Yeah, it was quite fast, but had cooling issues. His radiator was already like 3" thick, which I thought was too thick, and recommended going to a thinner radiator; instead he went to a 4" thick rad and the issues got worse.
Y'see, when you do that, you add to thermal capacity in the short term, but once everything is equalized, that back-half thickness of the radiator is being "cooled" by air that's probably within a few degrees of the water temp going in to the radiator to begin with... And since that back half of the radiator isn't doing much, if any cooling, the water flowing through it mixes with the cooler water from the front and brings the overall temp back up. This leads to a feedback loop and ends with overheating. In order to overcome this limitation with thickness, the amount of air that would have to flow through the rad would be...impressive to say the least. It might actually have been fine over 100 MPH, but that doesn't help when you're in stop and go traffic.
A good example of this (while not an engine coolant radiator, the physics are essentially the same) is Subie TMICs; they use a water sprayer to help with cooling, because there's not enough frontal area to do the job (those intercoolers are THICK, if you've ever seen one). Even then, it's not enough long term to keep temps down without spraying a LOT of water.
And yes, the stock radiator is stupid thin. Mine started to leak so I replaced it with a Koyorad. It is only about 1/2" thicker, so its not huge. Still it's definitely thicker than stock. You almost make it sound like its not worth the trouble. But since it is thicker, it does have some additional thermal dissipation, yes? And since it has higher capacity that means higher heat capacity yes?
I have a Koyorad as well; IIRC, the stock is 1/2" thick while the Koyo is 3/4" thick. The extra thickness allows for slightly more coolant capacity (and the coolant in the radiator will stay there a touch longer due to slower overall flow through a larger core), but the aluminum endtanks (vs plastic) help drop a tiny amount of heat themselves, and the increased thickness is arguably better at higher speeds/airflows.
Keep in mind, there are people running around with stock radiators and making 400-600HP without cooling issues.