No, 'the fact', is not redundant.
After all, 'the fact that' does serve an important purpose. The truth is, owing indiscriminate use this phrase has acquired a notoriety, so much so we now tend to actually read through it as if it is a mere filler.
This reflects the fact that his family is very supportive.
only references to the 'fact' and not to the actual subject 'his family is...'. The difference may be subtle. In other words, the writer is not especially concerned with what the fact itself is here, but its existence, and its confirmation by another supporting entity.
Consider a situation where the author has discussed some issues that throw up a question as to whether the family was supportive. He follows up with other statements that strongly suggest that the family indeed was.
He is now presenting no new information such as that the family is supportive. Rather, he is showing how the foregoing goes to establish a certain fact.
Remember, though, that most writers may not bother to see this difference and use the phrase the fact that carefully.
As for grammar, I'm afraid your contention that it is necessary in terms of structure is unfounded. The sentence is grammatical and makes sense either way.