The shimmy seems to be a very evasive animal. Some have it and can't seem to get rid of it regardless of what is done, some have it initially but have gotten rid of it with balancing, alignment, new tires, etc., and others never experience it. Were the S-Types you drove 2002's or earlier? I've only driven 2002's, and none have had any shimmy at all. Mine is a 4.0 Sport, and before I bought it I drove a different 4.0 Sport, a 4.0 w/o sport, and a 3.0 w/o sport. Every one of them was smooth as glass up to ~90mph (which is the fastest I personally would ever go). Before I took delivery of mine, I had them balance and align the wheels to be sure they were perfect from the start.
I'm certainly no expert, but this is my opinion: I don't think that there is a design flaw in the cars, per se, but that they are very sensitive to things like alignment, balancing, etc. I bought a set of 16" take-off wheels from a 2000 S-Type which I had fitted with Michelin Pilot Alpins for the winter. When I first drove the car at highway speeds after putting them on, I finally experienced the shimmy everyone talks about. I gave them about 250 miles to get worn in (I have a 96 mile per day commute), and when it got no better I went back to the tire shop and had them re-balance the wheels. They did so and said a couple of the wheels were "ever so slightly" out of balance (he mentioned one was only 1/2 gram out). He told me that on the majority of cars, this would be within tolerance and you would never feel anything, but that with some higher precision cars, being even a tiny bit out is noticeable. I'm happy to say that after the re-balance, the car is smooth as glass again.
My suspicion that Jag decided that the P4000's were the culprit is an educated guess based on my experience with my car, and what others have said about the Pirellis. I've never personally driven on them so I can't say from experience. Several have posted that the Pirellis tend to wear unevenly, and I suspect that if this is the case, it could cause the balance to become off enough to create a noticeable shimmy.
You mentioned that you have the MXV4's on your kitty. I had them on my Lexus GS300 (which the Jag replaced) and still have them on my wife's Avalon. I personally think they're super tires, and had no trouble with them on either car. Again, I'm no expert, I'm a gas passer by trade (anesthesiologist), but what I would do is ask the tire shop to check the balance again with the caveat that they need to make them as near to perfect as humanly possible. Their definition of "OK" may be allowing the tires to remain a fraction out of balance, which would be acceptable for other cars, but not temperamental kitties. If that doesn't do it, alignment has also been implicated as an issue, so you may want to have that checked as well (again insisting on VERY small error tolerance).
Hope this helps. Check out the JagTalk board I mentioned above. There are folks there who really know their stuff.