Maybe this is why they are not too worried about the "hacking up" of the wreckage?
I tend to agree. Before the tail was recovered, there was an official statement that the rescuers first priority was finding the casualties, the second was the black boxes.
The main interest in the wreckage, as reported, was about the position of the control surfaces (or "tail flaps" as it was reported) which could give valuable information about the airplane's last flying moments. Based on that information, the interesting pieces would IMHO be the actuators and the THS jackscrew.
Presumably the rescuers also took enough underwater photos and videos to enable them to establish the pre-recovery condition of the wreckage. I'd assume that to be standard practice in this kind of salvage jobs.