We usually call these our "safe foods" and yes, it is a very real problem for many of us in the autistic community, specifically around nutritional deficiencies. In a similar vein, as a child I went several years just eating plain Cheerios. For a close friend it was chicken nuggets.
"letting me" is hardly how I'd put it. More like I would refuse to eat any and all food unless committed to a hospital stay. Eating Cheerios and taking supplements was a compromise for my exhausted parents and they were more at a loss of what else to do. I don't think you understand what living with someone with moderate to severe autism is like. It's not being a "willful child" it's an entirely different experience than what most parents are exposed to.
