Heat the olive oil in a large Dutch oven or pot over medium-high heat. Add the onion and bell peppers, season with salt and cook, stirring occasionally, until the vegetables are tender and lightly browned, about 10 minutes. Add the garlic and cook 1 minute. Add the chile powders, cumin, coriander, oregano and tomato paste. Cook, stirring, until the vegetables are coated in the spice mixture, about 2 minutes, adding a splash of water if the mixture is dry.
Add 2 cups water, the tomatoes, beans, bay leaves and 1 teaspoon salt to the pot. Bring to a boil, then reduce the heat to a simmer and stir in the squash and rutabaga. Partially cover and cook, stirring occasionally, until the vegetables are tender and the chili is slightly thickened, 30 to 40 minutes. (Add up to 1/2 cup water if the chili gets too thick.) Remove the bay leaves and season with salt. Serve with assorted toppings and cornbread.