So, my logic on this is pretty simple.
Coolant in the radiator with the thermostat closed will spend more time cooling down and therefore reach a lower temperature. When the thermostat opens you have a cooler charge of coolant to reduce the engine temperature.
Yes, more flow will cool better, not because the speed of the coolant itself, but the fact that a larger amount of it is providing cooling in a shorter time. But circulating hotter coolant reduces the advantage. If flow was the only concern then why wouldn't the manufacturer make a thermostat that flows more. Even when fully open they have a pretty small area for coolant to pass through. Along those same lines why have a thermostat at all? Why not just have a valve that opens once the vehicle is up to temperature and never closes again. On a typical engine that is running properly if you leave the radiator cap off and put a thermometer in the water neck you will be able to watch the cycling of the thermostat.
Obviously a larger radiator and/or more airflow across the radiator will increase it's ability to provide cooler coolant to the engine, that wasn't the question that the OP asked.
Many situations have been mentioned, but the OP is living in Phoenix and gave a specific example. 120 degree weather, Now in AZ this means dry air as well, it is reasonable to assume that the air conditioning would be running meaning you aren't pulling 120 degree air (by the way that is in the shade) through the radiator it will be much hotter because the A/C condenser is heating up the air before it gets to the radiator. If the coolant is (we'll say a higher number) 230 degrees coming out of the block and into a radiator and the coolant and the air temperature is going to be closer to 160 degrees because of the condenser, how much will it be able to cool passing through the radiator?
I get it higher flow is good - in the block, but higher flow rate through the radiator is not so good. The thermostat helps on that end.