Your first paragraph is correct. The second one is only correct if the radiator is not able to sufficiently cool the engine. If the radiator has enough cooling capacity for the engine, the thermostat will typically not fully open because return flow from the radiator. A lower temp thermostat is not meant to repair anything, and is not a proper repair of anything. A lower temp thermostat is meant cater to either modifying the engine to produce more power (creates more heat), or driving it at full throttle for long periods of time (creates more heat),
The point of the low temp stat is to get that coolant out to the radiator for temperature regulation at a lower temperature, thus keeping the whole engine at a lower temperature (provided the radiator has no problem keeping up with dissipating the heat). There a few reasons to want to do this:
-Cooler engine/combustion chambers less likely to knock/ping under high load
-Cooler engine does not have as large of a heat cycling range
-Cooler external engine parts and wiring lasts longer and takes less heat wear
-Cooler engine can generate more heat (make more power) without overheating
-It can also allow the engine to be run at full throttle (maximum heat produced) for a longer period of time without overheating
-Engine stays cooler when idling for long periods of time (only if fan is running at the same lower temp as well)
-helps to keep oil cooler
Drawbacks during normal driving/non-race conditions:
-Fuel economy suffers because combustion chambers are not preheated as designed for factory operating temp
-Emissions are also higher because the cooler engine is less likely to have a clean and full burn of the fuel
-Lower combustion chamber temp caters to forming carbon deposits
-Heat from the HVAC system will not be as hot
My street/road-track Jetta has a 160°F thermostat in it, down from 189°F stock. It has a tuned low power engine that is held near full throttle for lots of time during performance or race driving, and it’s a toy that sits around waiting for that kind of use. It doesn’t have HVAC, is only driven when it’s warm and dry, is given an “Italian tune up” every couple of times it’s driven to keep carbon deposits at bay, and rarely is used for commuting.