Base input, common emitter has the highest voltage
and power gain. Its inverting gain is also useful in specific instances. That's probably good reason to focus on it.
Emitter input, base to a fixed voltage ("common"), is not as often seen in switching circuits. The emitter must sink the full collector current, so you haven't really solved anything, and the collector output is noninverting so you haven't even begun to implement a logic function. (Presumably you could gang some together to make AND or OR gates, but these are not universal!)
Similarly, the power gain is lower. The signal source must deliver Pin ~ ΔVbe*Ic; this will be much less than Pout ~ ΔVce*Ic for a typical load, but hardly zero. Whereas ΔVbe*Ib can be approx. hFE times smaller, on top of the advantage in voltage gain.
It is relatively common in RF circuits, where the base acts as a shield between input and output -- Cce is much less than Ccb or Cbe are, and this allows more gain-bandwidth without running into stability problems. At high enough frequencies, the marginal drop in power gain may not be a concern anyway -- operation might be at such frequencies where the gain is already fairly low (due to capacitances and other limitations); you might as well go for the configuration which gives better performance in other respects.
Note that I'm giving reasons drawn from analog amplification. This is for good reason -- a transistor "switch" is only a "switch" in as much as we call it that. It still necessarily goes through its linear operating range, even if only briefly; and in doing so, it takes some amount of voltage and current or charge at its input (the ratio of which gives an impedance, roughly speaking, and the product of which gives power or energy), and that causes its output to traverse its range (which has similar units, and the same meaningful quotient and product). We might not be talking about small-signal gain and impedance, but rather the average taken across the end points; we can be much more sloppy with them, since we're going to be passing around min/max margins, but the units themselves don't magically change, we can still use them just the same.
The distinction between analog and digital, is hierarchical: digital is a subset of analog, in which we can make certain useful assumptions, and use them to quickly solve different kinds of problems (i.e., combinatorial and sequential logic). This is only possible as far as our base assumptions remain true; and when we forget them, nature has a way of gently (or not so much) reminding us of this, from time to time.
Point being -- this circuit more easily violates those assumptions, so it also makes a poorer example of a "switching" circuit, and hence why you might not see it so often in textbooks say. So, basically coming around to my second paragraph, but with more background; and again, it's perhaps a less universal starting point anyway, so most digital logic isn't very concerned about it.
So, that said -- you will perhaps be pleased to know that such circuits
do show up from time to time, in professional digital circuitry. Example:
https://www.nxp.com/docs/en/application-note/AN10441.pdfHere (Fig. 1), MOSFETs are used instead of BJTs, because the gate terminal draws no current; no resistor divider is needed for the base. It could be implemented just as well with a circuit like yours, it just takes extra parts.
Note that the body diodes (shown redundantly in the symbols, to make extra sure you're reminded of their presence!) allow current to flow from the left side to the right, when its voltage becomes higher. The supplies are shown 3.3V on the left and 5V on the right, for precisely this reason. If the supplies were reversed, D and S would have to be swapped. Such a circuit does not work as a general level translator (any voltage on left, any voltage on right), but when you have a fixed configuration, that's perfectly acceptable!
Tim
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