Ah yes, you have described my life. We have often said "This kid defies all laws of physics!" because he has BOUNDLESS energy and yet hardly ever seems to eat. The air-plant - he survives on air. The Ten Calorie kid. "What, three bites? Well, I guess it'll keep him alive until tomorrow morning at least, and we can try again." Mine also tends to not eat much at mealtimes, then suddenly he's STARVING and needs to snack on something RIGHT AWAY, and I don't have anything except maybe oyster crackers or applesauce cups.
So one thing I do is to have lots of healthy snacks on hand. He loves Larabars (those energy bars that are made from dates and nuts), organic yogurt, oat bran sesame sticks, apple slices, Kashi TLC crackers with peanut butter, applesauce, fresh fruit, HUMMUS with crackers or blue corn chips. That kind of thing.(It took a LONG time and a lot of exposure to get him interested in hummus, and he still only likes the Sabra Luscious Lemon one, but hummus is an extremely healthy food. So for a long time, I would set out hummus and pita chips on the dinner table to snack on while he's waiting for dinner. No pressure, I just put it there for anyone who was interested, and eventually he got curious and tried it out.) They're also big on presentation - they're more willing to eat cheeese dip than a cube of cheese, even a spinach dip more than spinach, more willing to eat fruit off a bamboo skewer than from a bowl.
Sometimes kids like to do "taste tests" if the item isn't too foreign. Like put out several cubes of cheese, or several different kinds of fruit, (make sure she likes at least one of them) and then rate which ones she likes.
They also like to help cook sometimes. There are lots of healthy muffin recipes and I sometimes make healthy muffins on the weekend. Or carrot raisin nut balls or oatmeal crunch balls or some other kid-friendly recipe like that.
I understand about have no time, (2 kids, full-time job, and other family crises that I won't delve into - let's just say I'm stretched thin.) I'll admit that I've changed a lot of my eating habits to accomodate my kids and make tortellini and ravioli a lot more than I did before the kids. When I make dishes that I know my kid loves, like black beans and rice, risotto, or split pea soup, (we're vegetarian) I make sure there is plenty of leftovers. Then if I make something for dinner that he doesn't care for, I can serve him something different in about 30 seconds. Things like PBJ, soup, chili, microwave scrambled eggs, leftovers, or making a quesadilla take practically no time to make.
If I'm making something that they are unfamiliar with, I encourage the older one to at least try the new thing, and sometimes he does with an open mind, sometimes he reluctantly agrees to try it but he's already decided he WON'T like it, so even if it were to taste like ice cream, he would say he hated it, which ends up being worse than not trying it at all, because he won't try it next time. So if he just isn't willing to try it, I no longer make an issue of it. He's more likely to try it the next time if I don't lecture him.
Also, Trader Joes has lots of kid-friendly and super convenient foods, especially in the freezer section. They have veggie burger patties, and risotto and pastas in freezer bags. 30-45 seconds in the microwave and that's it.
And don't forget places like Sweet Tomatoes and buffets where they can try things out at their own pace. I seem to recall when my oldest was 3, he would eat a lot of the pickled beets at Sweet Tomatoes, but no other vegetables.
Three is a notoriously finicky age - remember she's not being a pain, she's hardwired that way at a biological level, literally. Early humans couldn't always watch their 3 year olds every minute, and the ones who would eat any old thing were more likely to eat something dangerous. The toddlers that would only eat very very familar foods were more likely to survive. (I swear I'm not making up that last part - it's based on scientific research. See Meredith Small.)
Good luck!