For the things you mentioned, the vegan and gluten-free options are processed much more. Beef, for example, is arguably a “whole food.”
Gluten-free isn’t healthier unless you have specific conditions. Most people can handle gluten fine, and some vegan foods are primarily gluten (such as seitan).
Vegan isn’t inherently healthy, especially if your eating mostly processed foods. A primarily whole-food vegan diet is likely healthier and cheaper than most people’s diets though.