I was involved in a very long thread on the Leica forum, and there were several examples showing pushing in LR vs. pushing ISO in camera. One member measured the noise of ISO 160 pushed 3 stops in LR to actually be lower than using ISO 1250. I also posted example test pics unlabeled, and they were so similar that no one could definitively tell which was which.
Most newer cameras with Sony sensors behave the same way. Pushing ISO in camera is mostly only advantageous in getting a good jpeg/LCD preview with these cameras (Canon cameras and Nikon cameras without Sony sensors are usually the opposite.)
Granted, I only shoot in M mode, because I don't find A mode useful without an AEL button to lock exposure for more than one shot without recomposing, so I'm not using autoISO. I essentially shoot this camera as I would with two film speeds, but with the advantage of being able to push exposure differently for each picture, rather than needing to push the whole film role.
In using only 2 ISOs and sticking to one set WB (at least as a starting point,) it sounds like Godfrey and I shoot similarly.
p.s. For a long time, most MFDBs, which had similar Kodak sensor technology, relied on converter pushing. You could set higher ISOs, but it didn't actually raise the camera gain, but, rather, told the raw converter to boost exposure at import under the hood.