Different theoretical traditions use different terminology, so it's possible my answer contradicts whatever tradition you learned from (for example, I've never seen your a, b, c notation for inversion. I'll use 6 for first inversion and 6/4 for second).
At any rate, the textbooks I learned from and taught from (I've studied and taught exclusively in America fwiw) say that vi indeed can have tonic function. To the extent that tonic function is defined as a resolution of dominant function, vi in root position is considered to have tonic function. Obviously, it can't be ultimate tonic function, but it can substitute for the I chord in these contexts; some textbooks prefer to refer to it as tonic substitute function for that reason. By the way, your example of V-vi6 is generally not done in this style. It tends to sound like V–I with a wrong note. In fact some theorists don't even call the chord vi6 when it does seem to happen, preferring to hear it as a I with embellishment.
I agree to some extent with your hierarchy of finality, V–I6 is indeed weaker than V–I. However, I6/4 doesn't really have tonic function at all in the vast majority of circumstances. More often than not, what looks on the page like a I6/4 is actually just an embellishment of V—usually called a cadential 6/4 chord. Its function is dominant, or, rather, an embellished dominant function.
Does this make sense?
EDIT TO ADD:
Oops, forgot your other question. You're right that, because the leading tone is present (in major keys), the iii chord seems like it could have dominant function. However, in reality, this is virtually never the case. If you look at the notes of a iii chord (in C major, E-G-B) it shares just as many notes with I as it does with V. This makes it not particularly useful for either function, and this is why iii is actually quite rare in common-practice tonal music, with the exception of sequences or as a push toward the relative major in minor-key pieces. When in does occur in major keys outside of sequences, it tends to more often be used as a substitute for I6 or other embellishing roles. That being said, there are exceptional cases where you might find a iii used for dominant function, they're just very rare.