That's a great video. There have also been many controlled, scientific experiments as well.
https://www.technologyreview.com/2012/10/24/183079/verifying-a-vortex/The first experiment was done in 1962 by Ascher Shapiro where he was careful to document and control for other influences. Temperature and air currents were among the variables he accounted for.
He also used a plug, filled the pool in the opposite direction of the expected rotation, let it sit for 24 hours. Clearly the video makers used his experiment as a basis for theirs.
Shapiro found that water did indeed rotate consistently in the direction predicted, and another experiment in the southern hemisphere showed the opposite effect. After his paper was published, numerous scientists performed their own experiments which confirmed his results. Lucky for us, it was a popular experiment for a while so there was a lot of separate experiments all around the world to confirm it.
After checking the TFES Wiki I see it references the same magazine article I found and highlights the difficulties in performing such experiments.
Two of the studies were published in Nature, one for the northern hemisphere and one for the southern.
Bath-Tub Vortex, Ascher H. Shapiro, Nature 196, 1080-1081
The Bath-Tub Vortex in the Southern Hemisphere, Lloyd M. Trefethen, R. W. Bilger, P. T. Fink, R. E. Luxton & R. I. Tanner, Nature 207, 1084-1085.There was a similar experiment that also verified the Coriolis effect in water draining that was conducted way back in 1908 by O. Turmlïtz.
https://www.nature.com/articles/197480a0Bathtub draining is a pretty well researched area of science it seems.