Well, soup definitely should not be complex. Soup should be one of those things where you throw together your favorite meats and veggies into a broth, and enjoy! I've taken a peek at the recipe you posted, and I've listed some of the obvious changes one should make.
Pasta gone, beans gone, jarred spaghetti sauce replaced with a nice homemade sauce.
Simmer some tomato paste with your favorite Italian herbs and spices to make your own sauce.
Now to fill in the gaps left by the pasta and beans. How about chunks of eggplant? Eggplant is used a lot in Mediterranean cooking. It's mild, and would work like a soft potato substitute. I've never cooked with eggplant myself, but I assume you can throw in some diced eggplant near the end of the cooking process (don't want it to get too soggy!)
Or possibly cauliflower. Many people use cauliflower as a substitute for heavy starches such as rice and potatoes. It's also mild like eggplant, but not as fancy.
When I make homemade spaghetti sauce, I usually dice in veggies like mushrooms, green pepper, onion, and garlic. These could work good in a red sauce based soup.
Oh, and my favorite italian soup is definitely italian wedding soup. That soup is very different from Pasta E Fagioli, but one of the key ingredients is spinach. Diced spinach, simmered for a while in the broth, is surprisingly tasty! It would be an interesting experiment to try it in a red soup.
There aren't many problems in life that can't be solved by sleeping it off, or adding more butter.