No, it never would have happened unless he wanted to kill three people and paid to have it done. In his mind he wanted people dead and was willing to outsource it, at no point did he say "oh lol I bet you are totally kidding but I am going to pay you anyway."

he needed the "real world identity so I can threaten him with violence if he were to release any names... FriendlyChemist is a liability and I wouldn’t mind if he was executed... negotiating the logistical details of the murder, Ulbricht agreed to pay Redandwhite $150,000 in Bitcoins to kill FriendlyChemist. DPR paid Redandwhite, who later confirmed that he had received the payment and carried out the murder, and sent what appeared to be a photo of the dead victim to DPR. DPR replied that he had received the picture and deleted it,” and thanked Redandwhite for his “swift action.” [1]

Ulbricht initiated the exchange, and in other related reporting appears to defend himself only by noting that the murders didn't actually take place. As such it was a sting operation only in so far as Ulbricht, endeavoring to threaten &/or murder people, had those efforts intercepted by law enforcement before they could get to someone who would actually carry them out. It does not appear to be a case of entrapment where law enforcement was trying incite Ulbricht, urging him on to actions he would not otherwise have taken on his own. Because, again, he was the initiator.

[1] https://pdfserver.amlaw.com/nlj/ULBRICHT-ca2-20170531.pdf