They become doormats when their parents deal with everything for them, not the other way around.
That's a very broad brush you're painting with, and I disagree with it. A lot of bullying victims cope by withdrawing, avoiding social contact in unfamiliar situations and going to great lengths to avoid conflict.
Some become conditioned into being doormats for the rest of their lives, just to avoid conflict. IMO, it's a very bad thing.