I think it's a good idea, but there is some trade off As the tree grows, the vegetables will lose productivity as the tree takes up more light, and the plants will also be competing for resources in the soil. However, even if you are sacrificing some productivity from each individual plant, the total productivity from that area will be greater if you interplant, and the production will be more varied besides. I have a single Tam Kam asian persimmon, and it's still pretty small, so I can't say just how dense the shade will be in several years, but as of now, there's plenty of sun there to grow other plants. It's especially workable if you stick to cool weather vegetables, like lettuce and broccoli. I wouldn't expect to grow prize winning pumpkins or a bumper crop of tomatoes there. I'm in Eastern Pennsylvania, zone 6, so our climates are probably similar. Looking at my garden today, the buds are just swelling on the persimmon, but the chives are over a foot tall, and the volunteer miner's lettuce, which is great in salads, is already flowering. I have Good King Henry, a perennial leafy green for cooking, growing around the persimmon, and it already has a lot of growth too. Sorrel also does well in less than full sun, and is especially good in the spring. Just this spring I've planted ramps, which are another onion family plant with edible leaves and bulbs. They are a spring ephemeral, meaning they leaf out bloom, and the leaves die back before summer, like a daffodil does. Their natural habitat is under deciduous trees. If your priority is to grow the persimmon, absolutely underplant with vegetables. Even if they don't grow gangbusters, you'll get more veggies than if you don't underplant. If your priority is to grow heat and sun loving veggies in that spot, and you have no other place to grow them, then your decision is harder. I'd recommend reading about "forest gardening" for more ideas on this. The book "Edible Forest Gardens" by Dave Jacke is really good, and was written specifically for the Northeastern gardener.

While I am not sure that it originates from S.E. Asia, I would venture to guess that the common name is Indian Head ginger, Costus woodsonii. You might have better luck asking folks that are knowledgable about tropical plants.

I would like to point out there is a group on FB called Oklahoma Wildcrafting. It is related to this website where you can get info on edible plants. The thing about this group is they are very ethical and safety concious. I would recomend checking it out and working with someone knowledgable before foraging. I'm all for such a thread, in which I would include plantain, yarrow, purslane, lambsquarters and chickweed. But I'm a newby. I purchased seeds for plantain because I could not decide if I was seeing the right weed. I grow golden purslane, a domesticated purslane and love it. Jewels of Opar, most often grown for flowers is edible and grows in August. It is a relative of purslane. The leaves aren't as thick. An acceptable summer salad ingredient. Buckshorn plantain makes a crunchy winter salad ingredient, though the skinny leaves need something else with it. My husband is on blood thinners. A small cut bleeds a long time. Macerated yarrow will stop the bleading. I fed chickweed to the chicks, not realizing it was edible. I didn't know we had soap nut trees in Oklanoma, but we do. People buy them from India when they grow here. I hope to find one someday. Here's an article from Texas about them.

For Chestnut trees, you may have to pay for a part or full postage, depending on trade.