I'm generally a big fan of using weak inversion, owing to the fact that I studied analog design under professors whose backgrounds were power-constrained biosciences and power-constrained RF.
One problem with weak inversion is the poor f_t (transit frequency). For a given power budget and length, weak inversion requires the transistor to be made very wide. As a result, the transconductance may be high, the transconductance efficient per current may be high, but the transconductance per transistor area is poor.
This may or may not be a big issue for you depending on where the other poles are. This wasn't a huge issue for projects related to biosignals (dominant poles were usually associated with some later output stage or a dominant pole compensation), but caused some trouble for the RF related ones.
(1) What is the reason for so stridently wanting to avoid dipping into moderate or even weak inversion with the effect of our small signal on the gate? Given that it's well known that we get more transconductance in weak inversion than in strong inversion, it can't be that the gain is deleterious. Does it have something to do with linearity and distortion of the signal?
Output-referred linearity and distortion (i.e. how large of a swing we can drive into the output) is more a matter of headroom on the drain side of the transistor. If you imagine the simplest case of a common-emitter amplifier with a drain resistor, you get distortion when the transistor enters the triode mode of operation, or when it cuts off entirely. The allowable range of outputs depends mostly on the supply voltage and the inversion coefficient doesn't really matter.
Input-referred distortion (i.e. how large of an input swing is OK) is bound to happen regardless of the inversion coefficient that's being used, unless mitigated with the use of feedback which sets gain and linearizes the system. In this regard, a weak-inversion system with the same current is (superficially) worse just because it has a higher open-loop gain.
However, linear feedback around an amplifier or other structure will take the open loop gain and convert it to a lower-magnitude, more-predictable, and more-linear closed-loop gain. It's the same thing if you're adding feedback to an integrated amplifier in your VLSI design, or an op amp in an audio amplifier.
In practice, the closed-loop gain reflects the actual end-user need of the circuit, and isn't simply maximized. A strong-inversion amplifier with an open-loop gain of 1000 or a weak-inversion amplifier with an open-loop gain of 10000, will both provide around the same gain when using the same feedback network. The higher the open loop gain, the closer the closed loop gain is to the ideal gain set by the feedback.
What are the main problems with operating in weak inversion more generally? Given that it has higher transconductance, all else equal it seems we would want to operate there. I know some designs do, so I'm wondering what the difficulties are in doing so with all designs.
Another issue is that the gain being high isn't always great. A small Vth mismatch (or induced/coupled voltage or whatever) is much more deleterious in situations like current mirrors, where I don't need huge gain so long as both halves are properly matched and interdigitated.