Only the top action cams have stabilisation at 4K and those which have it, do it optically, by magnetically moving the sensor around.
On one I have, Sony X3000, I had to disable it (with epoxy resin!) because it was badly affected by the slightest vibration. So 4K is unstabilised, while 1080 is software-stabilised (using accelerometers to pick off the active part of the 4K-sized sensor).
I am going on a trip where I am taking an X1000 which has no stabilisation at all at 4K, and has the above software stabilisation at 1080. So it is similar to the X3000 now.
I want to render to 1080 anyway, so why shoot at 4K?
I need to do lens correction, which replicates pixels, and doing it 1080 to 1080 does blur the image.
And - this is the main question here - because the 4K source will be unstabilised, I want to try the stabiliser in Vegas 16. That must obviously be doing cropping. You cannot software-stabilise 4K to 4K.
I suppose one will have to try it and see how good the Vegas stabiliser is...
It is for a ski video, which is a notoriously difficult thing to stabilise. Most of the videos on social media are accordingly unwatchable.