When you turn on the blower
(or any electrical item on your vehicle which draws a large current for that matter) it places an additional electrical load on the alternator. This causes the alternator’s output voltage to drop momentarily (hence you notice the headlights dimming) before the control loop readjusts the voltage back to normal (headlights back to normal intensity). If you turn your AC on you should notice the same thing.
On the flip side, when you turn off an item which is drawing a high electrical load, you may see the intensity of the headlights momentarily increase momentarily as the alternator control loop adjusts for the reduced electrical load.
Both of these are normal.
If you’re seeing the headlights constantly fluctuating then there is an issue which should be investigated further.
More info needed. How old is the battery, have you checked the belt tension and tensioner lately. The heater motor pulls a good deal of electricity. If your battery is getting old or your serpentine belt is slipping a little your alternator will not put out its full potential of energy. The system can take a second to adjust the output to the new draw.
One more thing to consider: Do you have any way of knowing if your Pilot has ever been flooded? There are a lot of previously flooded vehicles out there being unscrupulously resold without this key fact being divulged, and sometimes their electrical problems don't surface until months or years later.
I'm just throwing that out there because you seem to be having all sorts of frustratingly enigmatic problems with your Pilot, as you've posted.
You should check the system voltage to determine if this is a problem or not. If you don't have a digital voltmeter, they are quite inexpensive on Amazon. Hook up the meter while a helper runs the Pilot. Voltage at idle should be around 14.4 volts. Below 14 volts at idle is a problem.
Have the helper turn on the headlights and fan. A brief dip is ok, but voltage should come back to normal quickly. If the system cannot hold around 14.4, then you probably have a weak alternator, and maybe an old battery as well.