Basic geotechnical engineering....influence of the footing is usually taken to a depth of 2B, with "B" being the footing width, and the influence taken at a 45 deg line from the footing edge downward.

I certainly, though, wouldn't dig out a couple of footing widths just to put in compacted engineered fill - unless it was required due to the natural ground not being suitable.

Absolutely you would. If the soil underlying the proposed foundation has insufficient strength or bearing properties that deems it unacceptable for supporting the foundation, it needs to be dug out and replaced with structural fill, that means you dig out as well as down (per above).

I hope that dBasement understood that I would not dig out good bearing soil just to put it on engineered fill. Of course, if the soil is poor, one could remove and replace. 2x might not be large enough for strip footings that normally have pressure bulbs to depths of 3.5 to 4x footing width. One might consider the use of digging out partially and using geogrids (see paper by Braja Das and Adel Hanna on Model tests for shallow strip foundation on granular soil in ASCE Special Publication No. 16, Special Topics in Foundations (1988).