Is it possible to increase available take-off distance for a seaplane by turning?
Yes when I was bush flying in Northern Ontario (C-185 on straight floats) I used turning takeoff runs from time to time when loaded and with limited space.  You start the turn once on the step.  Step turns are a normal skill on floats although it's not a good idea to do it on choppy water in high winds.
The guy in the video had insufficient space to take off even with the turning takeoff run and it looks like he couldn't be bother to head farther down the lake to get a better into wind run.  I wouldn't have done a turning takeoff like that in that sort of windy chop.  I would've started right at the far shoreline as far as I could go, and determined a go-no-go point using a landmark that was about 2/3rds of the total run.  If you're not airborne fairly quickly (by water takeoff standards) in that kind of wind and waves, something is really wrong or your airplane is grossly underpowered.  I suspect that guy was overweight.