Kedusha indeed consists of three (or four[2]) introductory statements, each of which leads into a pasuk (from Yeshayahu,[3] Yechezkel,[4] and Tehillim[5]) used by the angels and by us to sanctify HaShem’s Name. There is a machloket, and there are different practices, as to whether only the chazan says the introductory passages[6] or the tzibbur does as well.[7] The minhag is for the tzibbur to say the longer introductory passages of Shabbat/Yom Tov.[8]
The Bi’ur Halacha[9] deals with the dilemma of how the chazan should recite the p’sukim of Kedusha. On the one hand, the chazan should recite the p’sukim out loud, so that those still in the midst of Shemoneh Esrei can be yotzei with him.[10] However, if he waits for the tzibbur to finish reciting the p’sukim so that those who are still davening can hear him, he is reciting those p’sukim of Kedusha seemingly without a minyan. Some say that the chazan can accomplish both by starting each pasuk before the tzibbur finishes reciting it.[11] Others say that the fact that those still reciting Shemoneh Esrei[12] or the tzibbur as a whole[13] are listening to him makes him considered as though he is reciting the p’sukim with a minyan. One could even claim that the tzibbur’s present minhag to sing along actually creates a minyan for the parts they sing together, and although they just finished reciting those same words, one may answer Kedusha multiple times.[14]
Regarding weekdays, the chazan’s ideal practice is clear; he should start as the tzibbur is finishing, based on the above. Shabbat davening, however, is somewhat more complicated. Teshuvot V’Hanhagot[15] argues that since the relatively long introductory passages are not an essential part of Kedusha, if the chazan waits to start Kadosh and Baruch k’vod until the tzibbur is in the middle of the added next sections, he will no longer be able to connect himself to the tzibbur’s recitation of the p’sukim. For this reason, Teshuvot V’Hanhagot prefers that the chazan say Kadosh immediately, along with the tzibbur, loud enough for those in need to hear him. (Rav Moshe Feinstein,[16] in contrast, writes that the chazan should wait until the tzibbur finishes, because he assumes that it will be difficult for those still in the middle of Shemoneh Esrei to hear the chazan over multiple voices. It is possible, however, that he was referring specifically to the case of the weekday Kedusha.)
There is another practical possibility with certain advantages: One can start reciting (in the classic nusach[17]) Kadosh soon after the tzibbur, and then begin a tune for K’vodo in a way that will encourage the tzibbur to sing along with him before they recite K’vodo themselves the first time. In this fashion, the chazan attaches himself to the minyan, and the tzibbur does not end up repeating anything (certainly not a previous section, which your friend opposes). However, although these may be the best ideas from a purist perspective, our average shul is not the place where we would encourage purists to go against standard practice when there are reasonable alternatives that follow common practice.
The K’tzot HaShulchan[18] refers to a practice that on Shabbat the chazan says Kadosh (and Baruch K’vod) quietly along with the tzibbur and starts aloud only from K’vodo (and “Mimkomo…”). He surmises that this is based on the assumption that on Shabbat, everyone will have finished Shemoneh Esrei by the time of Kedusha, and they therefore will not need the chazan to be motzi them. If indeed no one needs to be yotzei, this works well, but this is often not the case in many present-day shuls.
There is enough to rely upon to allow the standard minhag of the chazan saying Kadosh after the tzibbur recites the next introduction, with or without the tzibbur singing along. It is true that from a purist perspective, it does not make sense for the tzibbur to sing Kadosh after introducing the next piece. However, while one may not speak during Kedusha,[19] it is difficult to consider the tzibbur repeating Kadosh along with the chazan a hefsek,[20] considering that all of Kedusha is interrelated. The idea that the tune should not make it seem as though Kadosh is part of the same unit as K’vodo is appropriate l’chatchila. Therefore, there is logic for you, as a talented and learned chazan, to start his repertoire of songs from K’vodo. (Indeed, there is something nice about “good-old nusach” being heard more often than it has recently been common.) However, we would not deem joint singing of Kadosh to be illegitimate, nor would we attribute the fact that many people do not understand Kedusha’s structure to the singing.