Dear R.,
I am a mom of 4 children. My youngest has some extra needs and had many eating/feeding issues over the years. I have studied for years reguarding this subject and have implented much of my research to help my son.
Our son was a poor eater in very much the same fashion as you describe. He craved the exact same foods (bananas, apple sauce, bread, and cookies). Like you, we worried that our son was not getting the proper nutrition and rightly so.
I know you said you are a vegetarian, but your son's diet is not vegetarian per se, it just doesn't contain meat. You may think that being a vegetarian is a healthy choice and that is okay for you, but in my studied opinion it is not sufficient for a growing child. I'm so sorry to sound critical here. Please know that there this no perfection here and we struggle with many things with our diet too.
Back to your child's diet-- the more of these foods he eats the more he will crave them and around and around you will go because he will continue to resist good foods (not that bananas are not good food,but the rest are not good or at least have minimal value).
Our son's cravings and preferences for these foods changed when we changed his diet. We incorporated bone broth and cooked veggies. We eliminated any grains (breads, etc.) and eliminated sugar (no cookies). We temporarily took out fruits. He can now pass up a cookie, no problem, and believe it or not his favorite food is his mixed veggies (carrots, cauliflower, onion, zucchini). He is now getting loads of necessary minerals (bone broth) and vitamins (veggies and fruits).
Your son's growth could be hindered from lack of proper nutrition, especially from lack of minerals.
If you would like to learn more about this aspect of nutrition, you can check www.westonaprice.org. Weston Price's research is very interesting and he did lots of amazing studies. I realize this may be such a shift from your current vegetarian diet, but I encourage you to be open-minded on account of the health of your little boy.
On another note in reference to the food having to be pureed-- Our son had this problem as well. Other than the food you mentioned above, our son would only eat stage 1 baby food until he was 4! We worked with him on this in several ways. One was to deal with sensory issues in the mouth (if you make it through this whole email and aren't turned off, you can ask me and I'll tell you how we did that). The second thing we did was to use a method of VERY gradually changing the consistency (this took a long time and I can also explain how to to this effectively). And third, the very diet itself of bone broth and veggies, etc (GAPS diet), because of the proper nutrition, improve the sensory issues.
I know this is a lot to absorb in one message, so please contact me to find out more. It would be nice to be able to help someone else, so the don't have to struggle with all that I struggled with. By the way, we started all this with our son less than one year ago. He now eats anything I put in front of him. He is growing like a weed and he is developing!
God bless you. I wish you the best. If I can be of help to you reguarding any issues regarding feeding your child, please feel free to contact me.
T. E.