The functions of iii and the ii are contextual according to Riemann and Dahlhaus. The first inversion of iii is dominant in the cadence. The iii is related to the i so like the vi is related to the iv. So – according to this table of this image – the vi is subdominant as well?
The core reason for this mistake is to no be not take the distinction of primary and secondary functions (Nebenfunktionen) into account .
Secondary functions are more contextual to the chord progression and voice leading; because these are the ones that characterize the primary functions. Key is their relationship to cadence. ii before the dominant is the subdominant relative in the cadence, but the dominant in the harmonic pendle of tonic and dominant ninth.
I understand where he wants to go trying to summarize for the sake of introduction , but the iii is a chord that tends to vi and to I: the 7´ scale degree in the fifth is definitely a leading tone.
Only after emancipation of tetradic harmony in jazz you have the ambiguity of iii as a tonic, because of the overuse of X7M chords.
Riemann explains more criterias for the interpretation of secondary functions in depth more or less at the start of the book that i don’t have with me here…
The core idea is to determine the importance of the major or minor third in the secondary chord, and the scale degrees of the voice. If the note behaves like a leading tone, the chord is of the dominant category. If this leading tone is neutralized, it is tonic. If any note repeats and becomes a tonic note in the cadence, the chord is well interpreted as a subdominant: we have then a typical plagal clause.
Functions are not degrees, and vice-versa. Here is the core spirit of the theory: it is not a theory of chords, but a theory of chordal relationships that become tonal relations.
*graphic from Video Game Aliance