Firstly, let's talk about "the meaning of the original translations". As the wikipedia article correctly notes, the Greek word in the Matthew version is pais (παῖς), which mainly gets translated "boy" or "servant", with variations on those two themes. The Luke text has doulos (δοῦλος), which normally refers to a slave, though American translations often avoid slave because the practice of slavery in American history inevitably schews the ability of American English readers to read 'slave' without thinking about the trans-Atlantic slave trade in US history.

Now, that said, it was a commonplace of Graeco-Roman life in the 1st century that a master would use slaves for sex, and indeed may have had a preferred slave for sex. And it was commonplace that it may be a male slave used for sex. So that is a possible reading of the text. But it is not the only reading of the text, and it is not the necessary reading of the text.

The wikipedia article is therefore quite right when it says "a number of highly controversial interpretations of the relationship" here. They are not a consensus view, they are highly controversial, and not just because it appears to run against 21st century Anglophone Christian values.

Now, among ancient commentators on the text (I'm combining Matthew and Luke here): Ambrose, Augustine, John Chrysostom, Maximus of Turin, Ephrem, Theodore of Mopsuestia, and Irenaeus, all refer to this text, and none of them suggest that it was a sex-slave. I suppose one might explain that away by suggesting that early Christians wanted to suppress homosexuality in the text, but personally I don't find that persuasive. Early Christians appear fairly uniform on matters of sexual morality, and while their language around homosexual activity is sometimes discrete, they aren't necessarily afraid to talk about it.

So, the 'how' question is that this story hasn't had significant impact on Christian discussions about homosexual relationships because from Antiquity onwards it has not been read in this light. Readings that suggest the slave is a sex-slave are all of recent vintage, and have not received broad acceptance.