How do I make my blended soup a more appealing colour?
Of course “appealing” is quite opinion-based, so let’s look at the problem in a slightly more neutral “how can I avoid the colors mixing when I blend the soup”.
In short, you can’t.
If you have a significant amount of green and red veggies, that is.
One of the appeals and key features of a classic minestrone are the colorful ingredients that give you a bright and versatile palette, almost a mosaic in a bowl. But let’s not argue with the question’s premise.
If you want to blend the whole soup, you need to stay in a limited range of the color wheel, which means for your soup everything that’s between red (as dictated by the tomato base) and yellow will be fine, as is white or translucent.  Stay away from green. Blueish ingredients are quite rare and not part of a minestrone, so we can ignore that. In your example recipe, the spinach is out.
Alternatively you can blend everything except “the green”. That means you need to either fish out all the green vegetables or cook them separately. That said, the spinach in your recipe will cook so quickly that you could even add it after the blending step. And of course you still get chunks in the purée.
If your desire to blend is not motivated by the desire for a chunk-free result but about thickening the soup, you could use alternative ways to thicken the soup. An easy way would be to scoop out a few of the starchy ingredients (e.g. the potatoes), mash them up and put them back. Or introduce an additional thickener, e.g. a starch slurry.