Why doesn't just a sense work as the subject in the following sentence? Why does it need in a sense?
A sense we were witnessing someone pushing boundaries and defying conventions.
You could have:
A sense of fear washed over him.
Yes, you can say:
✔ A sense of fear washed over him.
This is because a sense of fear is a noun phrase. What washed over him? A sense of fear. In other words:
[A sense of fear] washed over him.
[A wave] crashed into him.
[A bird] landed on the ground.
But your first sentence is also constructed in such a way that a sense is being used as a noun phrase—and that doesn't allow it to fit with how the sentence finishes.
✘ [A sense] we were witnessing someone pushing boundaries.
✘ [An apple] we were eating something out of the oven.
✘ [Some ideas] we were thinking nobody should work today.
The sentence can be rephrased in a couple of ways. Both are grammatical, although only the first actually makes sense:
We were witnessing [someone] pushing boundaries.
[A sense] we were witnessing was pushing boundaries.
In a sense, on the other hand, is actually an idiom that means something beyond the joining of the three individual words:
in a/one sense idiom
: in one way : from one point of view
// In one sense, he was correct.
So, when you add in to the start of a sense, you are changing it from a noun phrase to a dependent clause that is acting as either an adjectival phrase or adverbial phrase—depending on its context. (In your sentence, it's an adverbial phrase that modifies witnessing.)
In short, the sentence you are questioning can be (correctly) seen as follows:
✔ [In a sense,] we were witnessing someone pushing boundaries and defying conventions.
✔ [In one way,] we were witnessing someone pushing boundaries and defying conventions.
✔ [From one point of view,] we were witnessing someone pushing boundaries and defying conventions.