Why do many recipes instruct you to oil or grease the pan at the start of the recipe?

There are three good reasons to oil your pan beforehand:

1. Your pan needs to be ready as soon as your batter is. For cakes, the batter can fall apart while you are greasing the pan. This is especially true for cakes that have air whipped in, such as genoise, which can fall apart in a few minutes. Recipes containing baking soda can lose their fizz in this time too.
2. You can't forget to oil up the pan. In the heat of the moment, it's very easy to forget to apply lubrication before transferring the batter/dough; inevitably this wrecks the recipe.
3. Greasing the pan is part of baking mise en place and should be done before main cooking. Mise en place is a key principle of professional cooking, and means "everything in place." The idea is that ingredients should be prepped and ovens preheated, so you can do the actual cooking efficiently.

I see this most often in cake recipes. If you do whatever it is to make your cake rise (beat a batter containing whole egg; fold in egg whites; add baking soda and or powder to a quick bread) etc, then stop to grease and possibly flour the pan or little tins, the batter will start to fall, or will be too bubbly while you put it in the pan. So have the pans ready before you start.

You may find this instruction in a recipe where the batter or dough could easily sit around for 5 or 10 minutes while you prep the pan; it that case it's probably just habit on the part of the recipe author. But don't make that your first assumption. There are definitely times when it matters.

I have nothing to back this up, but I always assumed that you should grease the pan at the start, not because something happens with the grease, but because some recipes want you to act rather quickly. If you correctly mix a batter and then you have to grease your pan, can result in a different batter (heavy pieces drop down, whipped egg whites won't be as elevating etc.).

Again, this is my own speculation. I think that the problem in modern times is partly solved, by having oil spray that is sprayed in 2 seconds.

Or another reason: simply that you don't forget to do it.

I think you got this wrong "Is it to make sure the pan is cooled to room temperature by the time I get to it, in case it came from the dishwasher?". This sounds like you preheat the oven but leave the oiled pan outside.

You are supposed to preheat the pan and oil and the oven. If you start with hot oil, your food gets done quicker and absorbs less grease. The correct time is to wait until your oven has reached the temperature specified in the recipe; modern ovens have a temperature dial and signal you when the temperature is OK. Waiting longer (because your other ingredients aren't ready) isn't especially problematic, but there is no need for it.

Oven dials are often not exact. So you may want try to calibrate your oven with water (boils at 100°C) and sugar (caramelizes at roughly 190°C) or a roast thermometer.