I've used powdered milk paint a lot on softwood stuff--pine, doug fir, larch...sometimes on maple. I don't think of milk paint as hiding the grain. It does let grain show, but not like that new-fangled stain stuff. I like to use two or more colors and layer them, then with a LIGHT touch, sand/buff/ding corners, edges, and wear areas to expose the colors in the lower layers. I apply a tung oil type finish over the paint to give it some luster. Spend some time with some scraps and milk paint layers to see what you can do.
I keep a few colors on hand, red, blue, green, white (for adding to lighten colors), gold, and black. Look for the older tones in these colors. They shouldn't be saturated dark colors. It really is fun to work with. Dries fast and you can mix very small amounts so you don't have old opened cans of paint going expensively funky.
Nice little box. Isn't handtool work rewarding?