I think the point is that the cooking process starts well before boiling temp. Potatoes take a lot longer to cook than corn so you would want to take advantage of that fact. Where corn cooks a lot quicker you would want to have a shorter cooking time to preserve the fresh flavor and texture so already boiling water helps that.
You can put potatoes in water that’s already boiling. It’s fine.
I think the point is that the cooking process starts well before boiling temp. Potatoes take a lot longer to cook than corn so you would want to take advantage of that fact. Where corn cooks a lot quicker you would want to have a shorter cooking time to preserve the fresh flavor and texture so already boiling water helps that.
Same for carrots