It seems there are many sides to this coin. It doesn't take much research to determine that inflammation can be a part of the body's HEALING process - when tissue is injured the body sends in all kinds of tools to repair that tissue. Icing and anti-inflammatories have nearly the same effect - they both end up reducing the volume of fluids/repair tools. Much of our pain and discomfort associated with an injury are related to the swelling of the injured region.
So why should you ever ice or take anti-inflammatories?
* No one ever said the body's response is perfect - allergies are examples of over-reactions to minor problems. Inflammation and swelling in an area meant for repair of damaged tissues can cause problems for surrounding tissues. Sometimes we need to limit the body's response to prevent other issues from developing.
* Given the choice between being miserable for 5 days or being only moderately uncomfortable for 10 days, most would choose the latter. We can go on with our lives with moderate discomfort. Drugs and icing give us this option. The example given above regarding a cold is an excellent one. I remember this being discussed in a very basic physiology course in college - in one way or another the body tries to remove "bad" things from the body through urination, feces, exhaling, and even mucous. The professor intimated, if not directed suggested, that when we're sick we should drink more water to increase elimination through urination and only take enough drugs to enable us to get the rest we need through sleep.
* If you are a recreational runner who is independently wealthy, avoiding all drugs/icing and sitting on the couch until you're healed might be an option. For the rest of us who are either not recreational runners or who are not independently wealthy, we can't afford the time off from training or our jobs for every injury we experience. Taking anti-inflammatories or icing enable us to maintain some level of limited training and job responsibilities even through injuries. To the points mentioned earlier in this thread, we probably shouldn't use our pain threshold as the determining factor for how to limit our training while we ice or take drugs, especially pain killers.