The video is strictly classical in terms of ball and spoke models of the various components and their lock and key behaviour. Diffusion would be good enough for things to work but there may be a little bit of quantum mechanics helping things along as well.
It is probably no coincidence that a quantum Turing machine would have four rather than two "binary" states since in the quantum world a single comparison allows you to branch four ways. DNA and RNA may well be in effect a Turing machine of sorts exploiting quantum mechanics.
In a similar fashion the number of amino acids is suspiciously close to the number of ways you can branch in three quantum comparisons. If there is a role for quantum effects in life then it is in making these stages more efficient than they would be if purely classical dynamics applied.
It is diffusion limited but a little bit of quantum tunnelling may well help the right component to find its mark. I expect they are right about the speed it runs. I found the DNA copy process animation fascinating.
Unless and until we can find independent life on another planet (or this one in an isolated example of archaea) and see what its genome and biochemistry looks like we have no other novel examples to look at.