@Bok, a bit of reverse-engineering, I guess.
We know the two goals, which are to keep the pot warm when it is brewing tea, and to keep the leaves cool when it is not, and we have the tools. What actually started me on it was re-reading John Blofeld's old book.
When he described his gongfu set back in the 1980s, it had not just a tea tray for catching water, but also a tea-boat and a cup-plate; now why would someone need so many ways to catch rinsewater? And why is his tea boat deep while his cup-plate is shallow? Wouldn't it just be easier to have two cup-plates? Or one big cup-plate? Or to get rid of the boat and the plate and just do everything on the tea tray which catches all the water anyway? This got me to thinking about how tea rinsewater is used.
One thing that clicked was when Blofeld mentioned that his tea-boat was deep because he was taught to pour boiling water over the pot until it was almost half-submerged. Now why would someone do that? Well, the parts of the pot that are underwater are not suffering from evaporative cooling directly, so maybe a deep tea boat would do a better job at helping to warm the pot than a shallow cup-plate or a tea tray.
When Blofeld was making tea, he was taught that the tea rinsewater goes straight into the tea tray; so why then would some people pour it over the exterior of the pot, especially after having been poured into the cups, cooling it even further? There are the explanations about patina, yes, but the tea rinsewater will be cool, which is not helpful -- unless you pour it on the pot when it no longer needs to be hot, such as between steepings.