Good Shepherd Sunday was the week after that video was posted, and "The Lord's My Shepherd, I'll Not Want" was one of the hymns I sang in church. Between those two encounters, I found a few things to write about it.
The hymn is sung to the tune "Belmont," whose second musical phrase descends diatonically from D to F#:
The text there is "He makes me down to lie," so the lying down in the text is represented in the music.
The two other things I noticed are melismas that emphasize certain words in the text. The last line of the fourth verse is "And my cup overflows." "Overflows" is only three syllables when spoken, but in the hymn, it's sung to four (indicated in red):
This extra syllable illustrates the bounty that God provides. Like the cup that overflows, the word "overflows" itself has a surfeit of syllables.
The last two lines of the fifth verse (and the entire hymn) are "And in God's house forevermore / My dwelling place shall be." Like "overflows," "forevermore" (indicated in red) is sung with more syllables than it's pronounced with:
Instead of four syllables, it has five, and that extra syllable helps to represent that long period of time.