As to whether it was 'Goebbels complaining that his office would be receiving too many requests':
He had received personally some of these requests, true. But that tale is rather unlikely for the alleged volume. His office was overseeing quite different matters. The well known sentence "Who is a jew, I decide" is attributed to Hermann Göring. But in reality he did not have much competence or say in that matter.
The final decision for this arbitrariness officially lay solely on Hitler's desk, where he would judge about such matters mainly via photographs. Indeed in contrast to Göring Goebbels was in a few cases known to issue such orders that some people would get special exemptions. He did so a few times for useful looking cultural actors, independent from Hitler. That a few more 'got away' without much appeal to those three party members was then the result of some second tier officials who delayed a few procedures, while the official stance was still 'when in doubt, assume not Jew' (or those various degrees they created for classification of races). That stance diminished ever further as time went on.
(— Volker Koop: "Wer Jude ist, bestimme ich". "Ehrenarier" im Nationalsozialismus", Böhlau Verlag: Köln, 2014. doi)
The total number of such exemptions, granted until April 1943, with the help of a tiny minority of nazi officials granting really big favours or just sabotaging some procedures of asserting racial status, is guesstimated to be smaller than 1300 individuals altogether. All others suffered the well known fate.
