I don't think this is related to the double left turn operation, especially since the approaches to Campbell Ave are single lane left turns. There are several single-lane protected lefts along Speedway that operate with TOD phasing, as can be seen in this Street View image:
https://goo.gl/MGNdzr. The signs in the video are clearly unique in that "GREEN ARROW ONLY" signs are being used to restrict the movements to protected only, but it very well could be related to traffic levels. Or, in this case, a councillor who's getting involved where they shouldn't be. I would not be surprised if TDOT was forced to change the operation of this intersection by the councillor in question.
In particular, this sweeping generalisation really bothered me:
[Protected lefts are] are recognised internationally as the single safest way to manage a major intersection
The notion that protected-only phasing is always safer is total bollocks. Protected-only left turns naturally have less capacity than permissive lefts, and as a result of that, often have larger delays. As delays increase, drivers grow impatient and start doing things that drivers who were patient wouldn't do, like push yellow lights or follow too closely. What I see in Seattle (at protected lefts) quite often is drivers who turn left after the yellow arrow is expired (since they don't want to wait for another cycle), and nearly clip a pedestrian who entered the crosswalk, not expecting a driver to so blatantly ignore a red signal. I can't seem to find the study at the moment (I will find it and I will post it here soon), but one near me concluded that permissive lefts eventually prove safer because traffic levels improved, which as far as they could tell, reduced driver frustration, leading them to perform less risky maneuvers. I guess allowing drivers the liberty to turn in a gap is less risky than not!