Each corner piece needs to make contact with the core.

Consider when the top layer is rotated 45 degrees relative to the layer below. The corner is now about 0.707 units from the centre of the face, but the layer below only extends 0.5 units out from the centre.

If the corner pieces were smaller than about 0,207 units it would become disconnected from the cube as a whole.

So, any cube larger than (I believe) the 5 by 5 either needs to have bulging faces, or have the edge/corner pieces larger than the others.

Hmm, it looks like even a 7x7x7 doesn't need to have much. I guess it has a tiny amount larger edge and corner pieces though...

https://www.amazon.com/Shengshou-7x7x7-Cube-Puzzl-black/dp/B...

You are correct, the 7 x 7 x 7 is the minimum where a perfect cube of cubes is impossible. The Wikipedia article shows the issue with the corner cubes.

Apparently the 7-cube bulges slightly which helps to hide that the larger sized edges / corners

https://en.m.wikipedia.org/wiki/V-Cube_7

they wanted to make sure nothing could cut in