I did some further testing, and the TLDR; version is, using an open drain GPIO to switch the ground to turn on and off a device is a bad idea because the Rds(on) is too high.
The first scope screen shot shows the Vds of the open drain FET in yellow, and the green represents the current for the 3 color display running at 3 V. Unfortunately, my uCurrent seems to be on the fritz, because anything other 1mV/1mA range gives me way too low readings. I know the first spike is about 39mA because I had done some analysis on it earlier to figure out how much energy a screen update was using (I documented that here:
https://hackaday.io/project/134018-coin-cell-powered-temperaturehumidity-display/log/145333-improved-current-measurements ). Basically I was putting resistors in parallel across the uCurrent terminals to change the ranges to get a 1mV/10uA and 1mV/100uA range, to get the voltage level out of the noise so I could see it on the scope.
I wonder if that is a bad thing to do, because, even though I did this a couple of weeks ago, yesterday my uCurrent's lower ranges stopped working. Knowing that the first spike is 39mA, the 725 mV spike says that my 1mV/1uA range is now 18mV/mA. I have no idea if it is at all linear, but I don't think that is particularly important in this analysis. You can't really see it in the screen shot because the green trace is on top of the yellow, but Vds at the spike is about 1.5V, making the Rds(on) something around 38Ohms.
The second screen shot shows the 2 color display running at 3 V. Other than showing that a screen update on the 2 color display takes less than 5 seconds while the update on the 3 Color display takes more than 10, it also shows that during the time the displays are actually shuffling the particles around inside the display, the 2 color display is using current more consistently at a higher level than the real spiky 3 color display.
The third screen shot show the 3 Color display running at 2.4 V. Here you can see why the 3 color display continues to function at the lower voltages (I tested it down to 2V successfully). At the lower voltage, the peak currents are lower, but they are drawn for a longer time, so the Vds never gets out of hand.
The fourth screen show shows the 2 Color display running at 2.4V, and at the point where it starts to activate the display, it draws enough current to cause the Vds to reach a point where the controller on the display just kind of locks up.
If anyone disagrees with my assessment, please educate me. Thanks