Fraktur actually does use a partially different alphabet. For example it uses the Long s: https://en.wikipedia.org/wiki/Long_s and Half-r: https://en.wikipedia.org/wiki/R_rotunda

About the "bad argument", I can't argue with you, because I'm not the one arguing. You'll have to take it up with the author of these lines:

"In a hundred years, our language will be the European language. The nations of the east, the north and the west will, to communicate with us, learn our language. The prerequisite for this: The script called Gothic is replaced by the script we have called Latin so far"

(Besides, what's so strange about transposing Cyrillic to Latin? It happens all the time even today when people don't want to or can't switch keyboard layouts.)