An e-ink display allows me to get sucked into reading on my phone. It eliminates half my time-sinks but makes the other half a little nicer. A device with a bad screen would make my productivity go up overall, but I wouldn't be using the device to do any work.

So perhaps what's needed for productivity on the device itself is in fact a much bigger screen. Those "tri-fold" phones might be big enough, if you could somehow also fit a usable keyboard in your pocket along with the phone.