CITES is an organization which is generally regarded as having no real power
The organization may not have teeth (and that is up for some debate
but if you are trying to move a bird of prey across an international border, the paperwork and time it takes to get it organized is mind-blowing. That was the reason the Winnipeg Mississippi Kite chick never made it back south to be rehabbed and released down south where they are more common. To get the necessary paperwork in place was going to take so long that the chick would be imprinted on its handlers up here (because of its age/stage of development) so not a good candidate for re-release and finding a home for it down there was going to be more than difficult - they are specialized birds. On the other end of the spectrum, we had some folks who donated money to purchase a captive-bred chick for release in memory of their late wildlife biologist daughter. When the chick was a foster chick and I was luck enough to be able to catch a feather one day while it was preening at the Radisson. As a thank you, I framed a photo of the chick when it came in and later when it was fledging and wanted to include the feather. I had to do the same amount of paperwork for the feather as I would have had to do for the whole bird and I needed to make sure that the gift could go anywhere with the family. Took me the better part of the year. Which was fine, it should be just as difficult for parts of a CITES-listed species as it is for a live, healthy specimen otherwise there is an incentive to transport parts (eggs, feathers, skulls, whatever) which undermines the whole idea of being listed at all.
But then, that's just my opinion from my limited experiences.
As for delisting the birds nationally ... you should know that peregrines are already legally being taken from the wild in some states and provinces. Usually young birds (easier to train and not intefering a breeding pair) on migration north or south - usually south if I recall. This would loosen up the permitting process, but migratory birds like peregrines would still require permits to be transported across borders - international and inter-provincial (I assume inter-state as well but don't know that for sure).