Technically, the video probably isn't wrong; it doesn't claim to present an exact trisection. At the end it shows something like this:
\(\displaystyle \phi=88^{\circ}, \phi/3=29.33333...^{\circ}\)
\(\displaystyle <ZNB (\phi/3) = 29.29279...^{\circ}\)
So they are admitting that the construction does not really create a trisection. (That last line, appearing to call the resulting angle \(\displaystyle \phi/3\), is basically a lie, but the rest seems to be honest).
But then, who cares? The construction is perhaps relatively "accurate", but it is also not at all simple. There is no benefit in using a compass and straightedge construction to do something that is not exact; using a real compass, every step adds error. The only benefit in compass and straightedge construction is in theory -- that in principle it does exactly what it claims to do. It doesn't accomplish anything useful. An inexact construction, no matter how close it might be, is utterly worthless.
I should add that a construction isn't really a construction unless it is accompanied by a proof of what it does. There is no way to prove that this is "the most accurate method" (in what sense? compared to what?); and it can't be proved that it is exact, because it is not (and because no such construction can be).