The Capacitive Sensor library on a D1 Mini
Actually it looks like the answer was in the early 2017 thread all along. I switched to a 400k resistor and it works! (I've only verified this just for pins 5 -> 6, although I'd expect it works for any combination of pins not including D0 / D3 / D4 / D8.)

However, a caveat here -- it doesn't work well. With an Arduino and a ~6m resistor, you can sense a change in capacitance from several inches away, with a nice continuous increase in capacitance as you get closer. At least for me, the 400k resistor had very low operational range, only maybe 1/2 in. or so, and even then the differences were subtle. But increasing to a 600k resistor replicated the initial problem -- I just got the "-2" error message consistently. And going down to a 200k resistor just decreased the range as you'd expect. So there seems to be a very narrow operational range of resistance for the D1 Mini, with less utility than the Arduino.