So "picky" eating is often genetic and has very little to do with the parent or child. Frankly I think it's not a helpful term because it either implies blame on the parent for feeding incorrectly, or on the child for being obstinate. You are not to blame. Your child is not to blame. He's likely learning his independence and working through possible sensory issues. I will say that it is entirely up to you what your child eats. If you don't want him to eat chips and fish fingers, don't give it to him. You give him options you are comfortable with and let him decide which ones he wants. He may, he may not, and that's fine. Don't give him alternatives after the fact because it creates the expectation of "if I don't eat, then I get something better." Yes, it is tempting to just get them to eat something, anything at all, but this stage is quite normal for toddlers, and they survive. Let him decide if he wants to eat what you give him and if he doesn't, end the meal, try again later.
No substitutions. No exchanges or refunds. You either eat or you don't. No pressure, just options. Most kids will eat what's put in front of them eventually if they are hungry enough. If he's underweight, then it's a different story, and worth checking with your doctor, but as long as his growth is on track, roll with it. Remember that you set the menu, not him. If you let him decide what to eat you set that expectation for the foreseeable future and will be making alternate meals for him for a good long while. Lastly - the last bit you said about him walking around - I would try to put an end to that. It's a choking risk. I don't let my kid walk around with food, and if she eats something, I make sure she shows me her empty mouth before I let her walk away.
https://solidstarts.com/when-to-seek-help-picky-eating/ https://solidstarts.com/10-steps-to-prevent-or-reverse-picky-eating/ https://solidstarts.com/preventing-picky-eating/
Also forgot to add we all sit down at dinner time and eat together