If you'd ever tasted the difference between home grown, freshly dug potatoes and store ones you'd understand why they grow them!
Two of my favorites that would do very well for you there are mizuna and baby bok choi. Mizuna makes a large attractive mound and it's heavenly when stir-fried with some garlic and olive oil.
Other than clover, which I think might be a little too invasive for a vegetable garden, you could grow hairy vetch, winter rye (in the areas you won't be growing winter vegetables), or buckwheat for any bare spots in the warm season. Buckwheat makes strong smothering growth and lots of vegetative matter to be incorporated when they plow it (or that will die in the cold winter and cover the topsoil to prevent erosion). A friend plants daikon-type radishes in the late summer/fall to get deep tubers. They die in the winter and earthworms are attracted to eat out the dead rotting tubers during thaws. By spring planting time the soil is well-aerated from the worm tunnels and filled with worm casings. Mycorrhizal fungi spread via worm casings.
You *could* amend your soil with clay kitty litter - get the cheapest, unscented kind - usually it's straight out of the ground and the absorbent kinds are often very good kinds like bentonite. And any leaves you see raked and bagged in the fall can be added on top as well.