See Shulchan Aruch OC 109, 2. Magen Avraham sk 7:
פי' שיאמר עמו נקדש לדור ודור וכו' עד האל הקדוש ואח"כ יתפלל בפ"ע אם יוכל לסיים קודם שיגיע ש"ץ לסיום שומע תפלה [כ"מ בת"ה ור"ל חביב סי' ס"ו] ובת"צ לא יאמר עם הש"ץ עננו ברכה בפ"ע אלא בש"ת כיחיד (מט"מ מהרי"ל הגמ"נ):
[Someone who pray simultaneously with the Chazan] He have to say Nekaddesh, ledor Vaddor etc. ...
NOTE:
The OP author argued that this is not a proof, because the Chazan is in a process of Tefilat Chazara, so the yachid who is synchronizing with him has a special reason to say the words of the Chazara, ledor vaddor instead of ata kadosh. But when the chazan says his own tefila loudly, may be that he needs to say ata kadosh. (this is my understanding what Scimonster objects).
thanks to @Double AA I discovered siman 124. The Bet Yosef in OC 124, reports the minhag of thie tefila, in name of the Kol Bo, who called it "תפילה בקל רם", translated in Yidish by "הויכע" (high voice, hohe is high in German).
One of the statements of the Kolbo is:
והוא א"צ לחזור ולהתפלל בלחש דאם לאחרים מוציא לעצמו לא כל שכן :
He (the Shats) needs not to pray for himself afterward, murmuring. Since he can with the Chazara discharge other people (e.g. who don't know to pray), he can even more discharge himself.
We learn from the Kolbo a very obvious reasoning. The chazarat Hashats is intended for persons who don't know to pray. The ledor Vador nusach also!
For a quoted proof: The Baer Heytev sk 4 and the Mishna Berura sk 9 write explicitly the words ledor vaddor etc..