The goal of a fan is to reduce the internal temperature, not exhaust temperature.
Lower exhaust temperature can mean the inside is staying hot, which is a bad thing.
As long as it's not forcing heat to be diverted through other means, the only way it can lower exhaust temperature is by increasing airflow, which should lead to lower temperatures inside, as well. A given workload has a power utilization rate that's invariant of the cooling (assuming it's not thermal throttling). So, that heat must be dissipated and the only way you reduce the dowstream temperatures is by increasing the volume of air that the heat is transferred into.
Haven't you ever gotten into a cold car and noticed that the hottest air comes out of the heating vents at the lowest air speeds? Crank up the fan speed and the air coming out of the vents is now cooler. If you're trying to warm up frozen hands, it's best to keep the fan speed low. If you want to warm up the entire cabin, you might as well increase the fan speed a bit. In fact, my car does this automatically - while the engine is still cold, it keeps the fan speed down and only starts ramping it up as the engine starts to get warm.
But let me put it this way: when your thermal paste goes bad (which at least happened on ps4), the exhaust gets cooler, but the CPU gets hotter. Adding air doesn't help, because airflow was not the bottleneck. The fans rev up, but your console stays hot.
Even with poor thermal conductivity, the rate at which heat energy is transferred into the heatsink is proportional to the temperature difference. So, ramping up the fan will still have
some effect, even with degraded TIM since it's not as if its thermal conductivity goes to zero.
As I said above, the only way degraded TIM reduces the total amount of heat that needs to be dissipated is if the CPU gets so hot that it starts throttling.
That will reduce the amount of power being consumed, which directly determines how much heat is generated.