Headnote: I didn't watch the entire video, so I assume the summary of how DM orchestrates the combat round is italicized below. Please correct where wrong.
BLUF: the only opinions which matter in this regard are your own and your group's. The rules are guidelines (p.244: "Basic Combat Sequence: A framework by which the referee may coordinate battles. Each round of combat comprises two discrete phases. On each combat round, each side takes turns at each phase in order of initiative.")

However, if you want to further coordinate battles by stating that "each side takes turns at each segment (i.e., Melee, Missile, Magic and Movement, in that order) in order of initiative." then do so. I would not do so, as I prefer the more abstract notion of "you won initiative, take your action before the other side (within the two-phase constraint)." Segmenting moves combat more toward the tactical end of skirmish wargaming (which is fine, if that's your goal).