While I have kept all other variables constant and ONLY changed from square to staggered on the same car, the same day, the same roads and found no differences. You can't make that claim.
I disagree. I've done back-to-back tests and, yes, I can tell the difference.
Now, sure, if one drives on straight US roads with only the occasional four-way stop, the differences will be very minor. But take the car out on a nice winding country road and the differences are easily felt - most noticeably through the steering. These are the kinds of roads I drive every day - I live on the edge of the Cotswold Hills, here in England, and we have some great driving roads (shame about the road surfaces - but that's another matter).
I also know what good steering feel is, having had my first revelation driving a BMW 528 for many years. The Model S isn't in that league - too big and heavy - but my P90DL with its current tyre setup isn't too shoddy either.
My first calibration was a series of loan cars I had when my first car (P85) was frequently back at the service centre. The P85+ cars all stood out for their much more precise steering feel (and no, this wasn't just changing the steering setting to "Sport"). Over the past 3 years I've driven a whole range of otherwise identical cars, but with different wheels - all on the same roads - so I think I can definitively say that yes, it is the staggered wheels that makes the difference. Subtle, but very much real.
I'm in good company - Chris Porritt and Elon Musk both advocate this setup - and Tesla even started fitting staggered wheels to the non-performance models too. In fact, if you look at most high-end performance cars from Porsche, Mercedes, Aston Martin, etc., they all have staggered wheels - even the "all-wheel-drive" ones.
So, in summary: we can tell the difference, but I accept that for some drivers on less interesting roads, this may not be so noticeable.