Most noodle soup recipes that I see call for boiling and draining the noodles separately, then adding them to the broth already cooked. Is there any reason that I can't just add the dry noodles 10 minutes before the soup is done, and add a little extra water or broth to compensate for that the noodles take? Doing it this way would allow the noodles to take up some of the broth's flavor, and also save dirtying an extra pot and a colander.
similar to what julio said, the main reason is that typical wheat-based noodles release a lot of starch into the water, which changes the consistency of the soup. the starches can add a dirty colour to the water, but more than that, they can also thicken the soup undesirably (think of making a roux). Finally, if there are leftovers, the noodles can sometimes become completely water logged, making eating it the next day a bit of a soggy experience.
That said, with all these considerations there are times when you might add the noodles directly in-- non-starchy noodles (eg. like rice noodles) seem to do ok. also parboiling regular pasta before hand seems to help a lot with the starch and can still help you achieve the flavour integration you mentioned.
Some people prefer to cook them separately because when you do so, you can see the water changes color a little and they don't want that (mainly flour) in their soups.
Another advantage is that you can have your soup ready and just cook the noodles on demand. The noodles go to boiling water, the soup is just very hot, but not boiling. That's what restaurants do.
If those reasons are not important for you, do as most of us do and cook them together :)
With a cream-based soup the added starch adds desired thickness. It isn't worse for the soup than adding corn starch or extra flour, and color change would be indistinguishable. Same for a tomato-based soup, or an egg-flower soup.
I like robust soups in general, and I think throwing the water out in general is a terrible thing to do to a food product. Any soup that calls for discarding more nutrients and flavor from a product already 'enriched' out of the crop's natural value isn't one I'd be interested in cooking. Cooking flavor into the noodles beats wringing more out.
Well first of all, any sauce you make that'll have pasta in it, must have at least 1 to 2 tablespoons of pasta water in it, i.e. 1 to tbsp of the water you boiled the pasta with.
Second of all, it's all about the amount. If let's say you're adding a 1/2 cup of rigatoni or fussili then you're fine, add'er up; but be advised when reheating, pasta will break easily on you. If you're putting let's say skinny noodle type pasta, then don't put it from the start. What you can do is half boil the pasta in the normal water, drain it, and then add it to your soup 5 / 10 minutes before it's done.
I add corn starch to many soups because it makes a heartier soup and the broth sticks to the noddles better. But I still boil my noodles separately. Noodle/flour starch seems to have a taste, appearance and feel that I dont find appetizing. One thing about corn starch though is that it breaks down with the addition of salt. So thats why you find some some dishes, Chinese foods all watery when they were a thickened sauce before .. someone added more salt than the starch could handle. It not like anyone would go through the hassle of cooking noodles separately if it didnt make a substantial enough difference. Yet, this recipe Im looking at now puts the noodles directly in the pot and I had forgotten why .. now I recall. :)