5X and 6P cameras are indeed based on a large sensor, and choosing this characteristic appear to be indeed legitimately at the expense of OIS.
The IMX377 is a 1/2.3 type: 7.81mm diagonal
In comparison, the Galaxy S6/Edge/Edge+/Note 5, LG G4, is 1/2.6: 6.828 mm (in 16:9)
It's the same sensor diagonal size albeit in 4:3 than the 16:9 1/2.3 type in the Z1 to Z5, which don't have OIS either.
More sensor sizes in phones: https://en.m.wikipedia.org/wiki/Exmor
Note: 1/2.3 for 4:3 format is more surface area than 1/2.3 for 16:9 as this metric characterize the diagonal and not the surface area that captures light which evolves at the height x width (aka square)
Sony Global – Products IMX377
Sony Image Sensor for Camera