I pretty much disagree entirely with this.
First, it presumes that you understand the reasons why people get the tattoos they get. I read recently, it was either on here or on IG, a tattooer (I think for some reason that it was David Bruehl, but I'm not certain about that) talking about how there was a point in their career where they were tattooing a lot of yin yangs. The tattooer asked those clients what religion or philosophical belief that the yin and yang belonged to and none of them could give an even close to accurate answer, but that regardless of this, they were seeking to achieve a kind of balance in their lives. Horiyoshi III says in an interview in the book JBxH3 that a lot of people who get tattoos of Buddhist deities don't even know what they mean. They like the appearance or the name of the god, and maybe they learn more about it after. So even within what you describe as a "culture" people are getting tattoos that are outside of their realm of familiarity.
Second, it's useless to talk about culture as this monolithic thing. If you want to talk narrowly about western traditional tattoos, a lot of those are sailor tattoos or military tattoos, or are deeply rooted in those cultures and traditions, and if you want to talk about people "staying within their own culture" a lot of us wouldn't have the tattoos we have.
Third, at this point in time there are very few tattoo traditions that are "untouched" so to speak and aren't influenced by tattoo traditions from elsewhere, and I don't see this as in any way a bad thing. Shit, this isn't even a new thing. I read or heard somewhere...maybe it was from an interview with Horiyoshi III again?...that Japanese tattooing and this whole body compositions were originally influenced by Polynesian tattoos. You look at classic Western tattoos and you'll see all kinds of Asian influence there with dragons and what not--George Burchett went to Japan and observed the masters at work--not to mention that way that Sailor Jerry took that Japanese influence and brought it into Western tattooing with those incredible backpieces he did which have been so influential on the way we get tattooed now. What about Ed Hardy, who synthesised so many things into his tattooing? What about Thomas Hooper who does very European tattoos but is very open about how much he's influenced by Polynesian tattoos? I'm going to say that most of the exciting things that have happened in tattooing have happened precisely because people didn't "stay within their own culture".