I have studied the documentary broadcast here in the UK and I didn't hear any hint of the "a" but wasn't looking for "fr'a" - I still have the recording.

And I did re-run the whole sentence to see if I could determine the first word. Even allowing for the nature of clipped transmissions which was the character of the time, the first word is "It's". If it were "That's" I would have to hear "at's" and I don't. The other possibility (which I can't hear) is that the clipping was so bad that all we hear is "t's" making it sound more like a glottal stop. But not to my ears.

So "it's one small step ............"

Of course a documentary is not fact, and if you knew what they do - (arrogantly in the name of entertainment). But when I hear the audio in one long run I don't perceive any falsification unless NASA did it originally.

And I still say the synchronisation of audio and video is not automatically precise. We will never know.