Although the mystique of these instruments is legendary, and as much as I hate to suggest such things, I would think some kind of instrument foundation that serves as the caretaker of such instruments might be in order. The foundation would ensure that the instruments are carefully tracked, and would have the resources to keep the instruments in the public eye, and safe when they travel. Creations like the Stradivarius violins are cultural artifacts and works of art, and as noted elsewhere in this post deserve to remain available to the public. I realize that such a foundation could be subject to petty politics, and that deserving players might not always be allowed to use an instrument, but it seems better to me than to allow such things as happened to the Lipinski to occur (dropping from sight for a decade). Such instruments are now so old that regular playing and maintenance are vital to their survival.
There are already several endowments that do just that.
You however underestimate the extreme ladder climbing prestigehounds in the arts preservation industry.