To reassure you op, age 2 is peak picky eater.
My dd was terrible at that age. So picky, ate few fruit and veg, picky about almost everything except plain pasta
Now age 5 will eat:
Cucumber, cherry tomatoes, beetroot, yellow peppers, carrots, broccoli, peas, sweetcorn, spinach in sauces
Melon, mango, strawberry, apple, raspberry, grapes, bananas, blackberries
Plus a normal range of meals
First rule? They actually rarely need snacks. Drop them and you will find they will be properly hungry for meals and eat more.
Second rule: they don't need as much as you think. At this age mine used to over fill up on large portions of plain, palatable starches like sweetened cereal, weetabix or fruiry porridge at breakfast, then turn down a healthy more balanced lunch with veg & protein.