Caldo Tlalpeño Mexican Chicken Soup

Name: Caldo Tlalpeño (Mexican Chicken Soup)
Ingredients: 3 pounds bone-in chicken drumsticks and thighs, with skin; ¼ medium white onion plus 1 diced medium white onion, divided; 10 cups water, plus more as needed; 4 teaspoons kosher salt, divided; 2 ears corn, husked; 1 pound russet potatoes, cut into 1-inch pieces; 2 medium carrots, cut into 1 inch pieces; 2 medium chayote squash or yellow summer squash, cut into 1-inch pieces; 2 cups green beans (1 1/2-inch pieces); 1 medium zucchini, halved lengthwise and sliced into 1/2-inch pieces; 1 tablespoon corn oil; 2 large plum tomatoes, diced; 6 cremini (baby bella) mushrooms, sliced; 2 tablespoons chopped chipotles in adobo sauce; 15 epazote leaves, chopped (optional)