I'm not convinced that audio recording and transcription by a court reporting service are standard at this stage of an NTSB inquiry (maybe these are routine but the fact has eluled my experience). Also, the prelim report refers to an "accident".... which I don't think is the correct terminology for what occurred.
As to the privilege against self-incrimination, under the Fifth Amendment of the U.S. Constitution:
As a first point, while politicos might have given "taking the Fifth" a poor reputation, the Constitutional privilege against self-incrimination is a very significant right (just for the record).
The privilege generally applies to criminal proceedings. The NTSB process at this time is far from criminal in nature. But..... the privilege against self-incriminatiin may be asserted by a witness in any type of proceeding if there is reasonable possibility testimony by that witness would incriminate the witness in future criminal proceedings. And a subpoena does not disarm (it doesn't over-ride) the privilege.
My moderate-to-strong inclination here is to rely on what I understand is the very high level of effectiveness and professional competencies of the labor organization which represents American's pilots. They may, for example, sense a search for fault-finding and scapegoats is looming. Rather than investigation of the incident giving fair or equal time and consideration to the way ATCOs communicate at JFK, and/or to the set of changed procedures the F/O reportedly was required to review, the APA reps and legal counsel may see Board or other highly visible authorities trying to tag the AA crew as wet laundry needing to be cycled, or hung out as the case may be, to dry.
With FAA reauthorization pending soon in Congress, and with FAA part of a Cabinet agency headed by a political appointee whose aspirations, not to say ambitions, for higher office are well-known, it strikes me as not surprising at all that labor organization reps and legal counsel are playing this match very cautiously. Not surprised at all. (I know aviators on the line wear suit coats as part of their uniforms, not jackets; still, back in the City of Chicago, County of Cook, State of Illinois, we would refer to this cautiousness as, "My clients are not gonna wear the jacket on this.")
EDIT: the above written and posted prior to seeing APA statement in post immediately preceding mine; no further comment, and this post's content not edited.