More likely he chose not to allow you to post links to other youtube videos. Dave's video is correct and sums it up well, presenting both sides but concluding that for practical purposes current does flow through a capacitor.
It's pointless arguing this with someone who has already made their mind up to look at it from a purely physical viewpoint and choosing to ignore the bigger picture. Simply put, from a purely physical point of view, ELECTRONS DO NOT FLOW through a capacitor. That's what his YouTube video shows you, and it is correct. There is nothing wrong in his video. However, from a practical point of view, for an EE, there is a current flow both into and out from a capacitor. The electric current flow into the capacitor causes an electric field to be built-up inside the capacitor, which causes a displacement current across the plates of the capacitor, and this displacement current in turn causes an electric current to flow out of the capacitor.
So for a capacitor, we have: electric_current -> displacement_current -> electric_current
This displacement current is caused by energy stored in the electric field between the plates of the capacitor.
Similarly, current flow into a primary side of a transformer (inductor) causes a magnetic field to be created, which induces a current to flow out of the secondary side. Current can flow "through" a transformer, but doesn't really flow "through" it directly. No one seems to have a problem with this.
For an inductor, we have: electric_current -> induced_current -> electric_current
The induced current is caused by energy stored in the magnetic field of the inductor.
And finally, I'll leave this mind-bender: with a diode (or any forward biased PN junction, for that matter)... we always say that current flows through a forward biased PN junction, because it's a simplification that works for us when designing circuits. However, if you get down into the physics of it, there is a depletion region at the PN junction that has no charge carriers whatsoever, and the majority current that flows across that depletion region is a displacement diffusion current. And yet again, no one ever argues that current cannot flow "through" a PN junction.
What makes capacitors so special that people have to take sides on this?