My cookie recipe calls for a starting temperature of 400, then a reduction to 350 10 minutes in and for the duration. My oven is big and well insulated, so it takes at least 30 minutes with the door closed to reduce to 350. Will partially baked cookies come out okay sitting on the counter that whole time? Maybe I should just start it at 350 and be done with it?
Unless stated otherwise, reducing the temperature does not mean taking the food out of the oven, but simply adjusting the temperature setting.
If you need to reduce the temperature quickly (usually just an emergency measure if the temperature was accidentally too high and the food is about to burn), just leave the door open for a bit, a few minutes should suffice.
After 10 minutes the bulk of the cookie dough will have temperature significantly enough below 400°F that enough heat will get absorbed there to get the temperature down. Of course this holds mainly if, as you state, your cool-off times come from good oven insulation, not if they come from large heat capacity like when using a cast-iron stove.
Also you want to be working a full tray.
The answer comes from the nature of the temperature control in an oven and basic thermodynamics.
Ovens work by having a heating element that adds heat to the cooking volume. Once the desired temperature is reached, the element turns off. If the temperature drops (because heat was transferred out of the oven), the element comes back on, raises the temperature a bit and then turns off. This cycling of the element (on/off) keeps the temperature within an acceptable range.
When you heat up an oven, you are heating up the air in the oven, as well as the walls, etc. that make up the oven. Once heated, they take a while to cool down. When you open the oven, there's some exchange of the hot air in the oven with the cooler air of your kitchen. When you put a room temperature sheet of cookies into the oven, the overall temperature will drop. If you look, the heating element will likely come on as soon as you do this.
By having the initial temperature set to 400, you will (as others have pointed out) have an environment that will act to quickly rise the cookies and crisp both the bottoms and the tops of the cookies. By leaving it at 400 for 10 minutes, you are causing the element to quickly come back on, adding heat to the overall oven system, but specifically to the bottom of the cookie sheet.
10 minutes later, when you drop the temperature, you aren't causing the temperature to drop to 350 degrees. Instead, you are telling the oven "hey, the oven is hot enough to cook the cookies now, don't turn the element on for a while". Eventually, the element may come back on, but if your recipe is "10 minutes at 400, 10 minutes at 350", the element may never come back on.
The ambient heat of the hot oven will be enough to cook the cookies. In particular, by not having the element come back on, the bottom of the sheet will get no radiant heat transfer from the element, preventing the bottom of the cookies from burning.