See, the stories are generally what I love about them. The first has the best overall conceit, taken straight from the classic yakuza film "Battles Without Honor or Humanity," where the main character goes to prison for some time and gets out to find that things have gone to seed during his incarceration. Yakuza adds some things to that conceit, but the big part of it is there. I would also argue about whether killing off Yumi is actually a fridging, since the general idea there is that the killing happens solely to motivate the protagonist. Her death serves the greater point of the story of the first game, that the criminal world of the Yakuza poisons and kills all who touch it.
The sequels' stories all work to varying degrees. Yakuza 2 feels a little like they didn't know what direction to go, which is why you get stuff like a love interest for Kiryu, who ducks out of the series in the opening minutes of Yakuza 3 and is never heard from again. Yakuza 3 remains one of my favorites, as a lot of the side stuff involves Kiryu doing stuff for the kids in his orphanage and slowly getting embroiled in local Yakuza matters. The whole series is all a lot of macho crime man posturing, but it works because of Kiryu, who is this lone figure in the middle who actually lives by those ideas of honor and integrity. His version of macho is also surprisingly not toxic. (Generally, its a weird and long series and I am sure there are bad examples.) They are pulp stories, but they are largely good and fun pulp.