I would have started with just letting him eat when he wanted.
I really think that it would have worked, if you had stuck with it. I think you have gone in the opposite direction by allowing your son to have books, his calculator, puzzles, etc.-- and you've even read to him! This is active distraction from eating.
Without being judgmental, and as a layperson, I suspect that this is largely or wholly a training problem (although I certainly do believe that some children tend naturally to be less robust eaters, though those same children would respond to training). Children simply do not need stimulation other than their food during the eating process, and it may distract them. (How many children in the world would skip a meal to play video games, as a simple example? Are their internal appetite mechanisms broken, or have they been trained badly, through passivity or otherwise?) I would start by making mealtimes just about eating. I would also see a doctor to check whether there is something wrong with the functioning of the appestat in all of your children, if you continue to think that they will consistently refuse to eat when there is a lack of distracting stimuli at the dinner table.
In terms of combating pickiness, I have read and believe that you may have to introduce a new food just once or many times, depending on the little one and other training factors at work. So one thing you can try is repeatedly presenting a new food and providing genuine encouragement to eat it, which of course may include you eating the new food in front of them. If you or your mate are picky eaters, you must not display those behaviors in front of the children.
In training kids to be non-picky eaters, you should search out new and wacky foods for them to try, to enlarge the horizon of what they'll consider. Be positive and consistent. Make mealtimes just mealtimes. I think it will work, but it may take some time undoing their current habits.
My wife's side of the family is full of picky eaters, and my DS4 started exhibiting some similar traits until I realized he was picking it up from her. Today he's a ravenous little omnivore. The only thing in recent memory that he refused to eat was octopus in its own ink from a can, and that only after trying it-- and after I tried it I almost hurled myself.
