Good question. The main difference between Push/Pull (or Offset) and Extrude is in the handling of non-planar geometry. Extrude won't work at all on a cylindrical or spline face which has been split, while (if it worked at all), Offset would allow you to select a split portion of, say, a cylinder, and offset it, resulting in the introduction of side faces, and an offset of the split portion of the cylinder (that is, with a larger radius). See attached picture
We haven't yet gotten to that level of detail in our plans, to be honest. But, this is very valuable feedback - I assume that you would like the ability to control the shape of the side faces? Normal to surface is one option - are there other options that you would also find useful?
To give that option a bit more depth, we might specify the angle of the side wall relitively to the surface (point) from which it extends. This may get complex and create conflict as surfaces extend outwards so a warning dialouge may be needed to inform user, or it may only allow the user to offset by a limited amount.
Square or triangular (or other) offset may allow to set a different angle for each side face.
Offset might also include a twist option or an arc option.