Chicken soup from scratch is best made from scratch chicken stock; and chicken stock is best made with wings, backs, carcasses, etc., rather than a whole chicken.
What you're talking about is using the reserved poaching liquid from (wait for it) poaching a chicken, along with the reserved carcass to make a stock -- and from that stock (which is not rich as a purpose made stock would be), make a soup.
Don't take all those caveats as criticisms -- they aren't really. It's just the way it is. What you're doing is a quick soup that can still be very good; and for lack of a better description we might as well call it Chinese style chicken soup because it's a very common way of making chicken soup in Chinese-American homes. The thing of it is, no meal is a meal without tong (soup), so let's smile and go forth bravely, heads high.
There are two keys to making chicken soup. Start with the idea that a chicken is a nearly, but not quite, perfect thing. One key to making soup is to get rid of all the undesirable stuff. The other key is to not do anything to mess it up.
You can break your 3# chicken into serving pieces if you like, or leave it whole. Remove any visible pieces of fat and reserve for another purpose or discard.
Submerge the chicken in a kettle in about two quarts of water. No salt, no vegetables, no nothing. Just chicken and water.
And heat. Turn the flame to high, and bring the water to a hot simmer. If it happens to boil for a minute, no problem. But that's the only time you can get away with it. Turn the heat to medium-low so that the water is it a simmer.
Scum will rise to the surface after a couple of minutes. Skim the scum and discard it. After a few minutes more, more scum will rise. Skim and discard again. Allow a few more minutes, and if more scum appears get rid of it.
Once the chicken goes five minutes without producing scum, add your "aromatics." Those are one onion, one stalk of celery (with leaves), and one carrot -- roughly chopped. If you like, you can roast the aromatics in a hot oven or saute until they get a little color. The precooking will make everything taste better, but isn't absolutely necessary.
Also add a dozen peppercorns and 1/2 a bay leaf.
Simmer until the chicken is poached -- about thirty minutes after it first hit the simmer. Leave the aromatics, bay leaf and peppercorn in the pot, while you remove the chicken. Take the meat from the bones. Return all the bones and the skin to the kettle, and continue to simmer for another hour.
Taste a piece of carrot. It will have all of the flavor cooked out of it, just like the rest of the aromatics and the bones. Since they have nothing left to offer, strain the aromatics from the liquid and allow it to cool slightly. Meanwhile, clean the kettle.
When the liquid is cool enough to handle, and the fat has had enough time to float to the top, partially defat it (leave just a little) by skmming or using a "fat separator." No matter how good you are at skimming, you're guaranteed to leave enough fat, so just do your best. If you're using a separator, go for about 90% clean.
Congratulations! You've got yourself a stock.
Return the strained, defatted stock to the clean kettle and simmer uncovered in order to reduce and intensify the flavor. You may use a bouquet garni if you like.
When the soup tastes good, add whatever fresh vegetables you like, and a little salt. (To my mind, it's the salt which turns it from a stock to a soup.) Simmer until the aromatics are crisp-tender. Remove the bouquet garni and adjust the salt so the soup is properly seasoned.
Return as much of the poached chicken to the soup as you like; you may also add some cooked rice, noodles and/or dumplings. Simmer another minute or two until all is heated through.