If a thermostat's job is to keep the coolant in the radiator long enough to cool down before letting it back in, I can't see how a cooler stat will change things in this case.
No, a thermostat's job is to begin opening at the rated temp, then only open more if the temp continues to rise. It shoud regulate temperature between 5° and 15° above rated temp (20° under extreme loads). If it cannot, then the cooling system is not capable of cooling well enough. If it can, the cooling system is fine, and capable of staying 5-15° above rated temp. If it cannot maintain temp, it doesn't matter what the thermostat is or how open it is, as the cooling system is not capable of shedding enough heat. See what I mean? It's a cooling capacity test.
With hot outside air a radiator with more surface area or better conductive capability would hopefully be able to transfer the heat before hitting the block again. A cooler stat would let the coolant out of the block sooner, but if the heat can't be exchanged fast enough I'm still gonna run hot.
Exactly. So, if it still runs hot with a cooler T-stat, then you have a crippled system. Another indicator is the temp difference between the radiator inlet and outlet. A functioning radiator will be 10-20° cooler at the outlet at max temp. Any smaller spread means the radiator or airflow is insufficient to carry the heat away.
If the spread is good, then you're looking at a lack of circulation, or for something that is adding more heat than it should, such as ignition advance issues. Just for comparison, I have a puny stock original 17" 2-row 302/351 radiator in-front of my 351W/427 stroker with a Ford 7-blade clutch fan and shroud. It stays cool in summer idling or driving. While big radiators are nice and can't hurt, it may not be the actual problem(s).