perhaps there was a small misunderstanding: degree theory itself contains "a bit" of functional theory, because it does not speak of degrees as melodic entities (i.e., the scale degrees) but as bases for chords (that is, degrees of a hypothetical harmonized scale). In this case, do, re, and mi are not the first, second, and third degrees of a C major scale, but are elements of three possible chords. This "forces" one to take into consideration not only the bass but all the notes expressed at that moment (and to untangle this polyphonic complexity, one must necessarily use a theory of chords, which is absent from the figured bass ostinato). This also leads to the possibility that do, re, and mi do NOT even belong to the C major scale. It is not those notes that qualify the tonality, but the chord they belong to, the series of chords they belong to. This consideration actually leans in favor of functionalism, but compared to a theory where the bass "governs" and the other notes are found at random and senseless distances from it, at least degree theory tries to identify harmonic entities, i.e., chords.
I don't know if I explained myself clearly...