This study by Bush et al. has shown that rare birds are not the most traded birds. My master thesis was on the international bird trade and I found that, although there may not be much trading in rare birds, the scarcity of a bird species does drive the price up. Additionally, poachers may have up to two years warning before a species is up-listed to a level of higher protection by CITES, causing poachers to fill their boots before trade in a particular species becomes illegal.
Unfortunately, the same study found that the demand for reptiles and mammals was related to their rarity.
The removal of over 13,000 birds from the wild may be disconcerting given that some populations of rare species are down to a few hundred individuals. Yet, despite the high volume of bird sales, the traded species were one and a half times less likely to be red listed than off-market species.