You can see in the video that he does have desoldering braid, but has chosen not to use it. Sounds like our experience differs, but personally I find desoldering braid is seldom helpful for removing parts - it tends to leave behind just enough solder to keep it stuck down, but not enough that you have good thermal conductivity to reflow it with an iron. Without using a heat gun or skillet (which he mentions) I don't know of a better way. You have a lot of experience, so if you disagree I would very much like to learn something!