As Rabbi Yissochar Frand concludes on his shiur on the subject, "as we say in the kashrut world ... it's not recommended."
As Rabbi Rakeffet points out, just using the term in your native language makes a difference here. He mentions that when he first moved to Israel if someone called him a chamor ("donkey") he thought it was cute, but to a native speaker it's quite the insult! Similarly he had an American woman argue for non-marital relations, saying "it's no worse than pilegesh", to which he replied -- "oh so you want to be some man's concubine?" That made her pause.
More from Rabbi Frand: Probably the most serious proposal for reinstituting the Pilegesh in (relatively) modern times was Rabbi Yaakov Emden's. He was concerned with the case of Jewish men who would move to America to work, leaving a wife behind in Europe (with plans that eventually he'd make enough money to bring her over, or he'd return home, or something) ... and years pass and he's still living on his own in America -- and as the Rambam concludes his Laws of Prohibitions on Relations, well -- celibacy is hard! Such a man can't "marry" a woman in America, as doing so would violate the ban on polygamy. Instead Rabbi Emden suggested "concubinage", which would have been a marriage in every way but name. It would involve kiddushin and would require a Get to terminate; the only thing it would be missing is it would not require a ketubah contract -- and even that, Rabbi Emden advises that they draw up a similar contract for her financial protection.
Earlier in history, it was debated whether a pilegesh requires kiddushin to enter and a Get to absolve, but that was Rabbi Emden's conclusion. The medieval rabbis debated whether the Biblical commandment to have children necessarily required full marriage (with ketubah and all the obligations thereof), or if concubinage would have theoretically been an alternative. Some (Rambam if I'm not mistaken) were of the opinion this was only a legal allowance for anointed kings -- as the Torah makes clear, kings have some special rules. Ramban counters by observing that the Book of Judges ends with a story involving a man and his concubine -- and quite clearly, "in those days there was no king in Israel"! The closest this man could have been was a "judge", who had some king-like authority, but still, if so, where do you draw the line, asks Ramban? So a commoner can't have a concubine, but a judge can? What about, say, a president of a major jewish organization? A millionaire CEO? The president of the local synagogue? (Note: If you are a synagogue president and reading this, I am not condoning non-monogamy. This is a thought experiment contained within a theoretical discussion!)
Between all the halachic questions about how Pilegesh works, and for whom; the reluctance to mess with the way things have been done for 2000 years easily; and the general understanding that Judaism values traditional marriage (according to many, even if Pilegesh is an option, real marriage is a bigger mitzva), it's just not recommended today.
As I like to quote, in Spain in the early 1400s they did one better -- "well if the bachelors and idiots are going to do their business anyway, better that it be without sinning seriously" -- and some rabbis provided "official grace, and communal funds" towards all-Jewish houses-of-ill-repute. Rabbi Isaac Arama (in an essay on understanding Biblical Sodom) was furious about this practice; it's one thing for individuals to have their struggles, it's another for society to enshrine their shortcomings into official public policy. The "official grace" described by Rabbi Arama was interpreted by R' Chaim Ozer Grozinski as having them all use the mikvah; I've heard Rabbi Yehoshua Grunstein say it meant conducting a pilegesh arrangement every time a um, client, showed up -- which points to your question. (I do not know what Rabbi Grunstein's sources are on this interpretation of Arama's essay.)