Phugoid is too long period a motion, typically it's measured in minutes, this was over in a minute or so it seems.
A strong headwind would indeed have caused increased lift at the fixed AoA. But there should have been no difficulty whatsoever in pitching the nose down to maintain altitude, since if anything the increased airspeed would have provided more absolute tailplane/elevator pitch authority. So the comment about running out of pitch control seems to work against that.