I was a horribly finicky eater, as early as 3 years old too. My mother would leave sliced hard boiled eggs and strips of bologna on a shelf that I could easily see and reach. I always ate the egg whites, but left the yolks. I don't know why I was like that. I can enjoy almost any food today. If you implement a similar program, offer some variety. I like Mia's idea of colors and bite-sized pieces of good, nutritional fruits and vegetables. Experiment, and see what works best.
A simple psychological trick with kids is to offer then a choice between 2 things. Never ask "Would you like green beans?". Ask them "Would you like green beans or lima beans?" There is something about the choice that gives them a sense of control. All you want is for them to eat some kind of green vegetable.
As a kid, there were certain types of food I never wanted to try. I didn't like the way it looked, or I didn't like something about the consistency. Kids will eat when they are hungry, and there are still lots of food varieties that a 3-year-old hasn't yet seen, never mind tasted.
I'm sure you are aware that you should consult a physician over nutrition and weight issues. Also, if you are experiencing the same heat and humidity discomfort that is plaguing parts of the US and Europe, make sure to drink extra amounts of fluids - beyond the simple desire to drink something from casual thirst. We need to double down on our water consumption when heatwaves arrive. Eat healthy, be healthy.