But it seems like you only need the stagger and overlap because you’re using circles in the first place. Would it look worse if you just divided the rectangle into 6 squares without any gaps or overlap?

My thought exactly. The sampling circles only enable you to (awkwardly) solve a problem that was fabricated by using circles in the first place.

I wondered the same thing, but characters usually don't reach the edges, so I guess circles fit the average character better?