Frankly, they would not be able to make a GTA style game narratively speaking about yakuza in Japan. The reason why they have been able to have these yakuza stories in these games is because you technically always play as someone who has been outcast from the clan.
Why? What do yakuza do? What kind of crime do they engage in? Is it even crime?
The Japanese are rather touchy about a lot of topics. The yakuza are one of them, and the series already skirts the limits of polite society for bringing up said organized crime but also other things like orphans, that Japanese Koreans and Chinese exist, or that Japanese judicial system is fucked up (they specifically bring up the absurd conviction rates in Judgment).
As for what crime the real yakuza engage in, just about everything. Blackmail and protection rackets (this is something that's sort of a unique thing, the way they do it is either just obtaining compromising information OR the pseudo-legal way of just being or arranging nuisance like far-right megaphone vans; yakuza backdoor involvement with the Japanese far-right is also kind of a paradoxical thing given that most yakuza fall in some or multiple sorts of category that Japanese extremist nationalists would consider subhuman, like Koreans or burakumin or orphans), drug trade, human trafficking (though in regards to these two, they like to pretend they don't), pimping, goons-for-hire, money laundering, smuggling, squeezing cash through talent agencies, etc. Kyushu yakuza as of recent are basically almost a terrorist group too, and I dunno if the Yamaguchi-gumi split is still in open gang war phase (these two things are basically why in the past few years the authorities actually started cracking down on yakuza in Japan).
What sets them apart from your usual organized crime is that they're not exactly underground, since those pins and offices are actually a real thing, and bosses are semi-public figures as well. A big part in this was that for a pretty long time there was something of an unspoken understanding between the yakuza, the public, and the authorities, which was basically along the lines of refraining from petty street crime (this is one basis for Kiryu beating up muggers at the start of 0, this was something that you supposedly could get your ass kicked on yakuza turf at least back in the day), especially when it comes to tourists, and in general keeping their activities behind closed doors so everyone can pretend they don't exist or that they're just "chivalrious organizations". Before the aforementioned split and the formation of the new Kyushu group, yakuza in general did a lot of PR work too to keep this status quo up, ie providing disaster relief after the Fukushima earthquake.