For example, if I cook some veggies in the oven at 450F and then need to cook steaks at 250F, is there any point in letting the oven cool to 250F? My expectation is that the amount of heat in the oven when it's at 250F vs when it's at 450F is negligible for something like a steak and that the significance of the difference in temperature setting is the amount of heat the oven continues to add to maintain the temperature. Is that the case? If so, are there other foods where it would be better to let the oven cool first? 
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. 