Respectfully, SaturnV, I'd say the priority starts about 15-20 minutes prior, to see what the crew were discussing and doing as they approached the time where things began to go wrong. Without the stage being set, some very wrong assumptions may be made regarding how and why crew responded as they did ... response being more along the lines of your timeframe. That allows the first run through of the synched "crisis period" to dispense with a variety of "they did what? why? how did that get into the equation" questions ... and this means fewer errors in various utterances coming out of BEA.
Regarding conflict of interest and the BEA.
It's there, the question is in what degree. It may be small, or large, and if it stays close to small, would be low risk to a quality investigative finding.