The C and the D are indeed fretted by the first finger at the 5th fret, but the C# and E are fingered above it, so those are the notes that sound. The reason the barre is effective is that the notes on the 1rst (A) and 2nd (E) strings will sound -- which would be difficult to do and still get the 6th string (A) note at the same time, unless you did the thumb-over-top.
But note, only a few seconds after he demonstrates the barre chord, he demonstrates the preferred R37 voicing and uses that for the remainder of the video.
The point of using only the 1,7,3 with the 2, 3, and 4th fingers on the 6 4 and 3rd strings is not to have to use the barre at all and so not to double the octave or include the fifth. Just use the pure root and guide tones.