This question is discussed in Chevel Nachalato 8:13, where a number of contemporary Rabbis' approaches are presented.
R. Avigdor Nebenzahl writes that he knows of no good answer to the question, but does note some hints to the shtei halechem, such as Chabad's text of ושני שעירים לכפר as part of musaf.
R. Ya'akov Epstein suggests that there is a hesitance to include any reference to sin on Shavu'ot (based on Yerushalmi Rosh HaShanah 4:8). If we were to mention the shtei halechem we would also necessarily need to mention the sin-offering that accompanied it; therefore we refrain from mentioning it explicitly at all.
He also quotes a Da'at Zekeinim miBa'alei HaTosafot (Bamidbar 28:25) who write that the phrase בהקריבכם מנחה חדשה which you note in your question is a reference to the shtei halechem as well as its accompanying korbanot.
Finally, R. Ya'akov Ariel argues that we only mention in our tefillot korbanot that are brought due to the festival itself, but not korbanot brought because of a different reason which happen to always occur on Yom Tov (such as the 'omer, the shtei halechem and certain korbanot brought on Yom Kippur).