に can be omitted precisely because you would naturally expect it in that position.
When it's obvious from the context that 何も isn't the subject of the sentence (with an implicit が), it must be the target of the change that would be normally marked by に when the verb is なる. So you can expect the listener to fill in the gap with に and understand the sentence as you mean it. In this particular sentence, the subject must be already known to be a person (or expected to be one because of たい) and this puts 何も into the second most likely slot for the verb なる even without に. With a transitive verb, an unmarked 何も would be directly put into the slot for a direct object (with an implicit を), but that slot is obviously not applicable for an intransitive verb like なる.
Some might still say 何もなりたくない is informal and it should be 何にもなりたくない, though. I feel the omission of this に tends to be accepted partly because 何にも with three /n/ sounds is particularly costly to produce.
どこも行かない sounds totally natural. When the verb is 行く, the destination is the only logical choice for どこも, which denotes a place and isn't expected to be a subject.
誰も会わない is ambiguous. It can be either the subject (with an implicit が) or the target of meeting (with an implicit に). The first choice would be the former.