Yeah, the head bobbing in the video seems to indicate that the bobbing mechanism is not just for counterbalancing without a tail, as has been supposed in the past; it has been previously imagined that theropods with bony tails (such as all non-avians) wouldn’t have needed to bob their heads, as birds today do it to balance their wait as they walk.
The bobbing style definitely changes, though, when the tail is added; indicating that the style was altered by the loss of the tail.
Frankly, though, the reports I’ve seen on certain websites (here, for example) don’t seem to actually look into theropod anatomy; they focus solely on avian anatomy. Most reconstructions of theropod dinosaur movement have indicated that bobbing was a necessary consequence of the biomechanics of the head.
There’s also the thought that bird head bobbing is also due to the fact that they have extremely constrained eye movement within their sockets; they definitely have to move their heads a lot more in order to see details around them (you can notice this when observing any bird, really). It really isn’t testable whether or not theropods had this problem; this feature evolved to remove most of the musculature around the eye that is necessary for extensive eye movement, which allowed for birds to grow larger brains.
So, whether or not birds lost the eye musculature and then grew large brains, or started to grow larger brains and then had to lose the eye musculature to continue to do so, is a mystery. The smartest non-avian dinosaurs (like Troodon) were at least closest to avian intelligence range based solely on brain-to-body ratio (keep in mind that intelligence is complex and to evaluate animal intelligence solely on this in general is flawed; however, it is the size of the brain when compared to the animal’s skull that impacts eye musculature, and thus is what we are interested in here); so it’s possible that they had lost the eye musculature at least by Averaptora or Eumaniraptora, though its unsure why that would happen. Frankly, the loss of such a feature is not observable in fossil evidence.
I doubt, if such eye movement evolved in non-avian dinosaurs rather than in birds, that it happened before Maniraptoriformes. Tyrannosaurus has some of the best vision in non-avian dinosaurs, and it was binocular and actually quite similar to humans in terms of eye position. I doubt the aid of easily-moved eyeballs - especially with a head as large as Tyrannosaurus’ - would have been given up, given their keen visual acuity and large visual sensory centers in the brain. As for Maniraptoriformes below Eumaniraptora (things like Ornithomimosaurs, Therizinosaurs, and Alvarezsaurs) had very small brain-to-body ratios; I would be very surprised if they needed to lose the musculature at this stage.
So, head bobbing while movement is probably characteristic of all bipedal dinosaurs, at least of theropods; moving the head around in order to view surroundings, however, might not have been present in non-avians, and most likely did not evolve until Eumaniraptorans (which, I will remind everyone, is Dromaeosaurs, Troodontids, and of course, birds).