I would hazard a guess that the video is talking about the cap jewels, if they are not removed small traces of dirt could be left in the jewel that could contaminate your new oil thus speeding up the wear process. If the bit of dirt is the right size it could possibly even enter the pivot hole to help bind up the pivot so the watch wont run. It may seem like a pain in the butt but you realy do want everything spotless and it may seem a bit over the top to keep cleaned parts covered up until you need them but if you think about it there are always airborne dust particles waiting to settle on your freshly cleaned work just waiting to get into the oil and start the wear process that little bit faster.
Hope that helps?
Paul.