The other advantage of brazing is that the lower temperature of the brazing metal means there is usually no possibility of actually melting or distorting the pieces being joined, as there is in arc welding, nor the kind of instant contraction / "pulling" that makes it a bit of an art to maintain the relative positions of pieces being arc welded.
I'd have to disagree slightly that the depth of weld is limited to the depth of the weld pool, unless one means in each pass over the joint; one usually makes several passes over larger or deeper joints.
Or that welding inherently involves only fusing together the original metal of the parts being joined: only forge-welding does not involve the addition of new metal to the joint.
Brazed or gas welded joints are also generally much neater, and less obtrusive or likely to require treatment with a "welder's eraser".