Michaela,
We do the exact same things. I also give him food while he plays, which is also not recommended by experts. We feel he needs good food to grow and that we will focus on sitting at a table to eat later. Like when he is developmentally ready for me to have an uninterrupted half hour phone call. He can do so many things that are beyond milestones, so I think the eating issues are a bit of asynchronousity. His brain is chugging along so quickly now, it doesn't yet recognize the importance of stopping and eating.
At this point, I am going with the most important and immediate issue -- food as the basic building block to grow. We eat no junk food here. A lot (a lot!) of meat, fresh veggies and fruit, and occasionally whole wheat pasta. We go through 3 avocadoes a week. He hasn't gained a pound in over 6 months. I can't imagine how he would do if we didn't distract and feed him with books, coloring, and (when I am exhausted) the tv.
This wouldn't be needed for most kids, but I truly believe mine does not feel hunger. (Full disclosure: DH and I before having him had to remind ourselves to eat because we would often forget to eat because WE were busy with work or a project of some kind. We now are better about it because we make such an effort to feed DS and model better behavior.)
