I was hoping a native Japanese speaker would answer this question, but it doesn't seem like we have any at the moment so here's my best attempt.
Ranma speaks a bit like a yakuza, but that's not how it's supposed to be interpreted. He does speak very rudely at times, but he's much closer to an average delinquent's (ヤンキー or yankee in Japanese) speech pattern than a yakuza's. The two manners of speaking are somewhat similar, because both of them are rude and disrespectful towards authority, and also somewhat cocky, and indeed Ranma is supposed to come off as rude, cocky, and disrespectful to authority. However, there are definite differences. I'd probably call yakuza-speak a sub-dialect of yankee-speak, but it has a lot of specific properties which Ranma's speech doesn't.
Rather than describing both dialects here, it'll be easier to link to places that have already done so. This blog post describes some basics of yankee speech. Here are a few lessons on yakuza-speak. As you can see, there are similarities, but yakuza-speak goes a whole lot farther in a lot of ways and has its own vocabulary. For instance, yakuza-speak usually has rolled Rs and slurred vowels (rare in yankee-speak), while both have some vowel changes. The difference is actually pretty similar to English. In English, if someone swears a lot and insults everyone all the time, you wouldn't assume they were a gang member, but you'd probably expect gangsters to talk that way, and you probably also would find that person a bit unpleasant even if you didn't think of them as a gangster. In all cases I looked at, Ranma seems to be squarely in the delinquent category, but not really in the yakuza category.
In fact, there's somewhat of a spectrum of delinquents in Japanese culture, with some speaking very similar to yakuza, and others not really fitting into the yankee archetype at all. For instance, the delinquents in Beelzebub often do roll their Rs and slur their vowels, but don't use most of the yakuza vocabulary. Ranma's speech is definitely less aggressive and offensive than that, but still less respectful than how a normal person is expected to speak. On the other end, Clannad's Okazaki Tomoya is a delinquent, but mostly speaks like a normal guy, and doesn't fall into the yankee archetype at all. However, it would be wrong to construe any of these characters as yakuza members unless there's evidence elsewhere. In some cases the more thugish of the delinquents may even be in gangs, but yakuza is different from just ordinary gangs (a better analogy would be the mafia).
However, the show is over 20 years old, and the yankee archetype, which was very popular in the 1980s and early 1990s during a period when action and martial arts shows were very popular, is much less common now. Ordinary Japanese people may not be as accustomed to it as they were at that time, so there's somewhat more room for confusion now. Speech patterns have also evolved quite a bit in that time separating the two dialects more and pushing Ranma further on the delinquent spectrum. Also, when Ranma transforms to a girl, she still speaks the same way, even though women are expected to speak more respectfully in Japanese society, which could lead to further confusion. And finally, the two dialects are definitely close, so some confusion is possible even without all the other factors.
As such, the misunderstanding is understandable, but if a person watched the show from the beginning, they would definitely not get the impression that Ranma is somehow related to the yakuza. Someone listening to the particular clips from the song you showed could easily make that mistake, since Ranma is speaking in an especially angry tone (especially in his female form) at that time and also because there isn't very much dialogue in the first place to make the distinction.
Unfortunately, the only source I have for this other than the above links and number of years of studying Japanese is a 5-minute conversation with one of my Japanese professors about this subject, so it's entirely possible that I'm wrong about some of the above assertions. However, I do think that I'm correct that you aren't supposed to interpret Ranma as speaking like a member of the yakuza, just a particularly rude and cocky teenager.
It's also worth noting somewhere that the average Japanese person's perception of how yakuza speak is pretty different from how they actually speak. Most Japanese people only know the yakuza through yakuza movies, which use a fairly limited subset of yakuza-speech to be easily understandable while still sounding authentic.