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But you didn't know that an inverted transistor does it better, which was common knowledge back in the 1970s.
Obviously, you can bodge the saturation voltage down to zero, but the crucial point is that the offset voltage of a saturated inverted transistor is a lot less than that of the same transistor biased conventionally. I've no idea whether the bodge current for exactly zero offset would be more or less, but you start off with less offset to cancel out.
One more case where you weren't paying enough attention as an undergraduate.
I got an extended lesson in the technique back in 1974, when my boss at Kent Instruments in Luton had me put together a slow but cheap and accurate two-quadrant PWM-based multiplier, where we used two saturating switches in series to get a very low offset voltage when the swtiches were on. The first transistor was run conventionally, and sank the bulk of the current, while the second was run inverted to minimise the offset voltage. Slow, but 0.1% accurate.