Well I did say the new game has recap vids to keep you up to date, but it's not the same as seeing it as it happens when you play.
There was also a recap part in Yakuza 2.. where you can view story summary of the 1st game, but it leaves out some little parts like side missions that adds more depth to the character. Does the recap talk about Date's daughter or Kage's son? I'm just saying it's recommended to play the first 2, not necessary.
I personally love the little side stories, but if you can do without them and knowing just the general plot is fine for you, then it's not a big loss.