I don't think either of these statements are correct. Assuming all thermal losses are extracted via the case (rather than internal airflow via vents), then for a fixed thermal loss (say the power draw at idle) the case temperature is also going to be fixed (all the heat reaches the case, case temperature then depends on thermal resistance from case to ambient air. All heat loss must flow (like a current) through thermal resistances to ambient air.
The internal thermal resistances will cause internal heat generating components to be hotter than the case by an amount based on that resistance. Higher resistance will result in higher internal component temperature - not lower case temperatures.
A lower case temperature coming from a lower case to ambient resistance will result in all those internal temperatures reducing.
The equivalent electrical situation is a current source (representing a thermal heat flow (power)) flowing connected via various resistances to ground (ambient air)
So heat generation in the power chip having the resistances : Silcon to chip case - chip case to heatsink/spreader (if used) - Heatsink/spreader to case - case to ambient air.
The temperatures are equivalent to the voltages in the electrical circuit. If you reduce the last resistor in the chain, then the "case" voltage will reduce - and all the other voltages reduce by the same amount.
I might have been the person who "introduced" this concept that better heat transfer means a hotter amplifier case. It is not strictly true as there are some (important) nuances to it that need explanation.
If you have two devices of equal size and shape and generate the same amount of heat, at steady state, the average chassis temperature will have to be roughly the same. The reason is: At steady state, heat generated = heat dissipated.
Heat transfer to the surrounding is governed by the difference between the chassis surface temperature and the ambient air temperature. If the amounts of heat dissipated to the surrounding are the same, with amp chassis of similar similar dimensions, their surface temperatures must also be similar (when averaged over the chassis surfaces), regardless of how well or poor the heat conduction paths from the heat sources to the amp chassis are.
What a poor thermal conduction path will affect is the time response, i.e. how quickly the chassis temperature will rise. To demonstrate I'll show simulation results of both cases (good and poor thermal conduction paths to the chassis).
Only a quarter of the geometry needs to be modeled due to symmetry. The heat released by the amp chip is transferred to a heat spreader (or heat sink). The heat spreader is mounted to the chassis, with thermal interface materials (TIM) applied to fill the gaps between mating surfaces of the heat spreader and the chassis. The chassis is cooled by convection (convection heat transfer coefficent of 7.2 W/m²·°C and 25 °C ambient temperature). We'll look at the differences between effectively applied TIM and poorly applied or no TIM (50x higher thermal resistance).
In the first scenario, the amp chip released 22 W of heat for 300 seconds and then the heating stopped. Below is an animation of the 2 temperature charts. When thermal conduction was poor, we can see that the amp chip temperature rose quickly and to a level much higher than when conduction is good. However, the chassis temperature rose slower and the peak temperatures didn't reach as high as with the good heat conduction case. Therefore, in this scenario, the amp with the poorer heat conduction will have a cooler amp chassis.
In the second scenario, we kept the heat generation on continuously. We can see that for this scenario the eventual steady state chassis temperatures are the same for poor and good thermal conduction. It just took the poor conduction case longer to reach the same temperature. Notice that with poor conduction the amp chip temperature went off the chart. That means the amp would probably shut itself down or destroy itself.
Therefore, the more accurate statement is that for a system with high thermal conduction to the chassis (or heat sink), the chassis will heat up faster than one with poor heat conduction. If you have enough patience and keep the test running for a long time, the one with poor heat conduction will eventually catch up. But you may need to keep the test on for half an hour or longer.
Below are the temperature profiles with heating on for 300 seconds.