Good catch. I use several packets of gelatin for beef stew. Adds a lot to the dish without adding odd flavors.
Lots of good suggestions already, here are a few riffs on some of them:
I agree with the potato starch suggestion. I keep potato flour on hand or this purpose. Be careful, it thickens as it cools and you don’t need much!
My Dad has a trick where he puts a couple of chopped potatoes and carrots in cheesecloth and cooks them down in the pot. He’ll then blend them and add them back. I love the flavor of this, and it’s the most nutritious answer, but the orangey color of the soup is off-putting to guests.
If you have a micro-planer, you can just grab a small red potato, wash it well and grate it in. It will basically be a mush going in, and dissipate pretty quickly compared to a regular grater.
I do agree with the stock suggestion. I’m not a fan of broth in general though.
If you prefer a slippery texture then konjac is the answer as somebody said above.
Last suggestion, if you decide to use a roux, I suggest bringing it to a light nutty brown instead of putting it in white. This give a much richer flavor.
Good luck!
I really don’t understand the stock vs broth opinions here. They not really all that far apart, depending on how you make them. Stock generally has more bones, and little meat; broth has a good bunch of meat. Stock has better mouthfeel; broth has better flavor. I usually split the difference by making a broth with a bone-in chicken pieces. All the flavor of chicken, but still gels in the fridge. Though I suppose you can also call that a “meaty stock.”
Or are we using the terms differently? And as far as the stuff in boxes go, whatever it is they call “stock” isn’t stock to me. It doesn’t even gel.
Chicken soup should have garlic in it, if that’s your thing.
I’ve read that using soup (older) chicken and/or turkey will give you better flavor. Meat won’t be as tender, though. If you’re skilled with a knife or buy from a butcher, you can buy a whole tender chicken and remove the meat from the carcass, using the former for other dishes and the latter for your soup/stew.
Tapioca is a thickener that hasn’t been mentioned yet.
I like the technique of cooking whole vegetables in a soup/stew and puréeing them afterward, but I usually leave carrots out of the process. While puréed carrot probably adds to some dishes, I don’t like the results in most of what I cook. Maybe a tiny amount would be okay, but I find that one puréed carrot in a pot of 4-5 servings results in a pervasive sweet orange mush that turns beef stew or lentils, for example, into something else.
In case it’s helpful to someone, I’ll mention that thickening a stew or soup increases the likeliness of it sticking to the bottom of the pot and burning, so it’s best to thicken toward the end of the cooking process. This is especially true if you’re going to add chunks of raw potato (beef stew, for example).
Yeah, you used to be able (and in some places still can) buy older chickens specifically for the soup pot.
Woops! Yes, chicken or turkey gravy packets would be apropos.
I make a roast chicken on Sunday night. After everyone has eaten their fill, I strip the carcass, throw the meat in the fridge, and throw the bones, with whatever meat is left on them, in a pot with some carrots and onions and garlic and salt.
This goes on the stove until bedtime or thereabouts. I cool it in the fridge on a trivet. Next afternoon, I strain it, and it’s ready for the meat to be added back, along with more onions and carrots. The chicken soup it makes is one of my family’s favorite meals.
I’m certain that we all had a good ol’ fashioned dust-up several years ago about just what the hell stock and broth were, but I can’t remember if anything was decided. Probably just petered out into senile gibbering and puns.
Hopefully, we can at least agree that the stuff sold in boxes like “Swanson’s” isn’t stock in any meaningful way. It tastes fine as a base for many dishes and stews, but it ain’t stock.
I add some, “better than bullion” chicken broth paste as a flavor enhancer. Might get salty if you’re not careful, though.
Yeah, if your soup lacks a little oomph, don’t be afraid to use this, bullion cubes, or just straight-up MSG to give it a kick. The all have a form of it in there, anyway.
My recipe is similar, except in addition to carrots and onions, my stock includes celery, parsley, sage, rosemary, thyme, salt, whole pepper corns, and – on the excellent advice of a Jewish chef – a pinch of dill weed. No garlic for me. 
I leave the skins on the onions to give a beautiful golden color to the stock base.
I usually have the breasts left over from my roast chicken and put those aside to add back in after the stock is made and after stripping usable meat off the stock bones.
Once the stock is strained and the bones are picked, I simmer fresh carrots and celery in it, then add the diced chicken meat in last so it won’t go stringy and dry.
This makes enough soup base for a couple of generous batches. I freeze half for another day.
To the base, I can add more vegetables and noodles for a straight-up chicken noodle soup. But if I’m in the mood for a thickened soup…
I prefer a gently-browned roux to add for thickening. The browned butter adds such a lovely flavor. I like to add cooked rice and wild rice to the base, then boost the flavors with extra thyme and add some cream or whole milk. This makes a very flavorful thick, rich, creamy soup.
There are a million different ways to go with a soup base and there are a lot of great ideas in this thread.