Translation:I want to be a kind person but they do not allow me.
Why is "won't allow me" not acceptable? The meaning is essentially identical, and it's far more natural (which is why I keep getting it wrong unless I remember the less natural construction).
"Won't" is a contraction of "will not" , future tense. That may be reason it is not accepted.
Nope. "Will" has multiple senses. The one you mention covering the future, and the other which covers volition.
Trờ thành why not "want to become" which is my understanding of the phrase