Well, what you are trying now (making him sit at the table until he eats and going to bed without eating), does NOT work.
You tried.
3 years old is a hard age. Worse than 2 years old. But thankfully, once they are 4-5 years old, it gets better.
My son is also picky, but he is healthy and grows like a weed. I also give him vitamins. And, he will not eat if he is not hungry. If he is not hungry, we don't MAKE him eat, just because of what time it is on the clock. We rather he know his body's hunger cues. We don't force him to eat. We cue into him, and learned that he is a "grazer." Meaning he does not eat in big quantities at one sitting. He grazes throughout the day. This is him. My daughter on the other hand, will eat all in one sitting. No big deal.
When I was a child, my grade school Teacher was a witch. She FORCED all the kids in the cafeteria to eat ALL their food, or they could not leave the cafeteria. Needless to say, this traumatized some kids, made others hate her, made others cry, made others scared, and just made the whole lunch time miserable. Some kids would pretend to eat, and then spit it out when she was not looking, stuffing it in their milk cartons. She even held a ruler in her hand to slam down as a special effect of her "power" over us. One day I got her wrath. I told my Dad. My late Dad, invited her to dinner at our home. He watched her, just as she watched the kids in the cafeteria. When she finished her meal, my Dad told her that she did NOT eat ALL the food we fed her and until she does she CANNOT leave our home. She was aghast and furious. My Dad told her off, in a gallant way. He told her that that is what she does to the kids at school...and this is our home and she will follow HIS rules. My Dad sat there and did not flinch. He made his point. He made a formal complaint to the school and she was reprimanded. AND other parents complained about her too.
Now, is this what you would want your son to experience at school? What if a Teacher did that to YOUR son? Would you be pissed? I think so. Some would call it going to far... to make a child eat and "obey."
But that is my personal experience. In my family, we do not force our kids to eat. But they must sit at the table and try their best. Food "phases" comes and goes. But personally, I do NOT want to make food an issue.
To me and my Hubby, food/toileting/bedtime is not an event we want to "force" or manipulate our kids for. But we explain our 'rules' and are plain and honest in our expectations of them. Main thing is that they try their best. And, I don't expect my kids to eat like me.
Perhaps, let your son make his own plate at dinner time. My daughter likes to do that. Then she eats no problem. My son, we "let" him, within reason, pick what he wants to eat, from what I have prepared.
No, dinner time is not always "fun." But, it should be a nice comfortable NON-stressed out occasion. Otherwise, a child will HATE meal time/family time. THAT is worse, right?
Just as the kids in my grade school just HATED that Teacher... and hated when it was her rotation to "supervise" the cafeteria.
Dinner ideally, should be a time when the family and the kids can talk about their day and their concerns or anything. THAT should be the main thing. Making it 'hospitable.'
Well, just my thoughts, and experiences with that. My late Dad... really made an impact on me when he put my Teacher in her place for forcing the kids to eat. Bless him!
All the best,
Susan