Thanks to whit for the very insightful comments on the AD's point of view.
I strongly suspect that Patek has extensive and specific information about the grey market. In the past, I worked at a major manufacturing company and our products had an active grey market. We purchased some units and traced them via serial numbers. It turns out they had been originally sold to our biggest customer. So, our sales team had a "chat" with them. Their response was - yeah, so? They said they bought more than they needed to get to the next volume price break and resold the parts they didn't need. If we would just give them the volume price for fewer units then they would be happy to stop. Well, you can probably guess where this went.
Look at it a different way - from Patek's viewpoint. The grey market is another channel for their product. Just as the used market (e.g. the collector's market) is a channel. The grey market channel holds some amount of inventory (good to some extent as that expands the market) and it also creates more efficient distribution (without any of the costs associated with an AD). Yes, of course, they do recognize the frustration it creates for us collectors. And they do put some amount of control on the grey market. But they don't take steps to entirely eliminate it.
And, consider one more point. Patek doesn't operate in a vacuum. There are many other high end watch vendors. Would Patek want to be excluded from an important channel (the grey market) where everyone else is playing? I think not.