how could using money for its intended purpose lead to the abolition of money? Doesn't demonstrating its utility only reaffirm said utility, thereby guaranteeing its continued use?

That demonstration of utility doesn't guarantee money's continued use. The money abolitionist realises he lives in a system where use of money is necessary in order to effectively propagandise his anti-money message. It's possible that he uses money in propagandising and is ultimately successful in hastening the 'end of money'. You and I think he's making a mistake, but he's not necessarily contradicting himself.