I would say the one thing that comes to mind about processed foods that is pretty harmful is the glycemic index spikes from the processed sugars and can cause insulin resistance. Also the saturated fats are pretty high, especially in vegan alternatives.
You're not wrong. I was admittedly being a bit pedantic when I said "nutrients." Technically that's stuff like vitamin A, C, B, calcium, iron, etc.
Calorie density due to added sugars and saturated fats are a problem. Those foods are also hyperpalatable, making it super easy to quickly consume more than the body can handle. Short term it's no big deal, but it builds habits that become chronic health issues over a few years.
And I know all this intellectually and still sit here with a BMI, waist measurement and age all within a point of each other...