Should an oven be left to cool if cooking at a lower temperature?

This depends very strongly on what kind of oven you have. Since there are many, many variations in oven design, I'm going to go over three contrary options based on my own personal experience.

Gas oven, freestanding and thinly insulated: with an oven like this, you can turn the heat down to a lower temperature, put the new food in, and expect that the new food won't cook a lot differently than it would if you allowed 10-15minutes for the oven to cool. Gas ovens depend heavily on hot air convection from the gas flames to transfer heat to the food, and when you turn down the temperature, if the oven is not well-insulated, the temperature will fall almost as quickly as the food heats up. The only food I wouldn't do this with would be something which was very sensitive to rapid overcooking.

Electric oven, built-in and well-insulated: my current electric oven, on the other hand, is very well insulated and relies on radiant heat from the walls of the oven to cook foods. As such, if I turn it down from 450F to 250F, I can expect that foods inside the oven will still be exposed to heat higher than 300F for at least 1/2 hour. As such, if I need to drop the oven temperature for cooking reasons, I prop the oven door open.

High-end electric oven with exhaust fan: I've also had the chance to cook in higher quality ovens that contain an internal exhaust fan with an exhaust pipe to the outside. These ovens actively cool themselves off when you lower the temperature, so going from 450F to 250F would take less than 10 minutes, despite the thick insulation. With this oven, you just turn the heat down and put the food in.

The intention behind an oven is to keep as much heat for as long as possible. There are even luxury ovens which are intended to never be turned off, and always use a constant temperature; communal village ovens used to work on the same principle.

Modern ovens are rarely built that way, for several reasons. But still, you can expect (or hope) that the heat saved in your well-preheated oven is significant, and that "the amount of heat the oven continues to add to maintain the temperature" is the negligible one. So I would assume that the new dish will get heated much higher than per recipe.

From there on, it is a matter of your dish's temperature sensitivity. If you place a cheesecake or a piped meringue into your overheated oven, you are asking for trouble. But if you are, say, roasting a pumpkin, then you probably won't have much trouble from the initial high heat. For a steak, you will likely be finishing it on a pan anyway. Just use a thermometer to make sure you don't have too high a temperature gradient between the deep and superficial parts.

Conveniently, Quora already has a thread on this with a post calculating the amount of time it might take for an oven to cool from 400F to 250F, and they came up with 10.6 minutes (the decimal number might be unnecessary, as the time varies from oven to oven). How long does it take an oven to lower its temperature, for example from 400°F to 250°F?

So it might be a better idea to let the oven cool for a bit before using it.