So I'm trying to wrap my head around if larger image size may result in better ISO sensitivity (so images get less grainy)?
It actually tends to work the other way: larger photosites give better low-light performance because each site has more photons falling on it.
So if you have two sensors with the same physical dimensions, the one with the lower pixel count will generally yield a less noisy (but lower resolution) image. Or, if you have two sensors with the same pixel count but different physical dimensions (e.g. APS-C and full frame), the larger sensor will offer better low-light performance.
The numbers for those two models bears this out. Your NEX-6 offers ISO settings all the way up to 25600, while the NEX-7 only goes up to ISO 16000.