My 2016 olympus pen-f, which wasn't exactly ground-breaking technology when it came out, already had an electronic shutter that worked with the full resolution of the sensor.

It's a 20 MP micro 4/3 sensor - so 1/4 the surface area of "full-frame".

Just realized, the Nikon 1 we got for Christmas for my daughter seems to have one as well... Need to test that! Still, we look at a 10 MP sensor smaller in size than a DX. Heck, the more I look at the Nikon 1 series, the more I realize to ehat extend it served as a mirrorless test bed for Nikon.

Do you expect the physical size of the sensor to have a bigger influence on this than the resolution?

Depends on the sensor, doesn't it? The more dynamic range, the more data a sensor is collecting.