Dimensions
Super 16: 12.5 x 7.4 mm
16mm: 10.26 by 7.49 mm
2/3": 9.6 x 5.4 mm
Like 16mm
So, the depth of field of a 2/3" sensor is like 16mm, a respectable format. I recently watched Duane Hopwood (Super 16), and I thought that many of its shots were beautiful.
Approximating 35mm
According to this depth of field calculator, 16mm has the same depth of field as 35mm when you open 2 1/2 f-stops. So I guess that 2/3" at 2.8 will yield a respectable depth of field.
More flexible
Don't get me wrong. I love the depth of field of 35mm. But I think I would croak in a run-and-gun shoot. In between your independent films, you might want to shoot a sports event, documentary, etc. The deeper focus of 2/3" would relieve you -- even though focusing with 2/3" can still be difficult.
Lighter lens
At the same quality, a lens for 2/3" will be far lighter and smaller than 35mm. Yes, still lenses are fairly light, but, again, they won't suit run and gun. First, they do not keep focus when you zoom. The lightest 35mm "cine" zoom that I've seen is a 3x at 5 lb. Another (5x) weighed 10.
Here is a shot I took 10 years ago with an even smaller sensor, 1/2":
