Very interesting.
This argument does not work quite as well as hoped, but the Big Bang is also not a proof of infinity and at least rules out an infinitely old, infinitely large steady-state universe.
This would be the case if you make the assumption that the universe started with the Big Bang. However, the Big Bang is not the origin of the universe, but the earliest state that we have evidence of. It is like the highest point of a mountain, which is not the beginning of the mountain, but the place where it starts to descend. The Big Bang describes how the universe evolved from an incredibly dense and tiny point, but it doesn't account for the laws of physics and other things. More and more cosmologists are moving away from the idea that the Big Bang was an absolute beginning and consider it a sort of "first moment of time", as Sean Carroll puts it.
I would say that this argument reminds me of another argument that apologists use to defend the existence of God using science to make philosophical points, such as in the Kalam cosmological argument. William Lane Craig is one of the main defenders of that argument but he was was critiziced by top physicists and mathematicians such as Sean Carroll, Roger Penrose, Carlo Rovelli, Adrian Moore and others on the plausibility that the universe may be eternal and and therefore, actual infinities may exist. The documentary is on Youtube.
