Yeah the series is definitely a lot more than just about the yakuza and even though Kiryu has had to share the screentime, the series is about him and his almost legendary persona as the Dragon of Dojima.
I feel "Like a Dragon" would work well as a series relaunch, at this point the only people who buy Yakuza are pretty much those who are on message boards, always asking for localisation of the Yakuza series and I doubt renaming the series to Like a Dragon will confuse much, if any, of the Western fans.
Personally the three issues I have with the "Yakuza" name is the game's centre themselves around either on Kiryu or his legacy and not the yakuza themselves. The original game kept referring to him as a dragon because of his inhumane strength, the second had two "dragons" face off against each, the fourth game was who would be Kiryu's successor as Kamurocho's dragon and from that point he is always referred to as the legendary dragon (Even in Of the End) and the previous two games have had as
Secondly I feel it was a dumb attempt by SEGA America/Europe to try and make this the Japanese GTA, personally I feel they should have targeted the RPG crowd rather than anyone else and I definitely feel people would be more open to trying it if the series was called "Like a Dragon"
Finally Yakuza sounds far too serious for what the series is. We've got a panty thief running around, Kiryu being chased by a transvestite, Doc Brown creating virtual reality simulators etc its a whacky series and Yakuza just makes it sounds too serious.