Take away: you need to do both to keep things cool!

So, you have read through various opinions and users reporting various results through doing various things in an attempt to reduce temperatures on their equipment. You wonder what you should try first?

There are two key elements to keeping things cool. The first one consists in removing the heat from the heating element (e.g. microchip) and passing it to other parts of the system. The second one consists in dissipating/removing heat from the system.

Physics fundamentals summarized in one sentence: both heat transfer and heat removal are proportional to the temperature difference between the two media between which heat is being transfer (e.g. microchip to heat sink, or heat sink to air).

For the heat to be taken away from the microchip, the microchip temperature has to be higher than that of the component it is transferring heat to, typically a heatsink. However, to ensure proper contact between the chips and the heatsinks, heat transfer media are commonly used. Commonly used media in electronics are thermal pastes/putty/pads and the likes. If in doubt that these are essential, note that air thermal conductivity is about 100 times lower than that of low-end thermal pads, and about 700 times lower than that of Kritical thermal pads. In other words, you cannot rely on stagnant air to remove heat. Good thermal media will ensure that you give a chance to your heatsink to do its job.

For the heat to be removed from the overall system, we typically rely on heatsink (basically a simplified heat exchanger). Key characteristics required for a good heatsink are (a) high conductivity, (b) high specific heat,  (c) very large surface area exposed to air/water, (d) favorable air/water flow through it. When built-in heatsinks are insufficient, adding copper or aluminum based heatsink modules to the hot elements, and forcing more air through the system are sure ways to see a reduction in temperature...that is if heat is properly conducted away from your microchips.

What happens if one or the other is neglected?

With poor heat removal/dissipation, the entire system and all of its components will keep heating up until the temperature increase enhances the heat dissipation process enough to achieve a stable state...from a thermal perspective, likely not from a system stability perspective. Even with highly efficient heat transfer away from the microchips, it will only be a matter of time before the chips is getting very hot.

With poor heat transfer away from the microchip, the microchip will keep heating up until the temperature increase enhances the heat transfer process enough to stop the temperature increase. Even with high efficient heat removal from the overall system, the microchip temperature will likely to reach uncomfortable levels.

By now you should have gotten the basic idea. You should not focus solely on one end of the problem as both heat transfer and heat dissipation are needed to keep electronics cool.