Is there sufficient interest in having Anglican Chant added as a category?

Good idea.

Or would something broader be useful, to take in other music of the Anglican tradition?

Personally, I think that Anglican Chant is a liturgical music genre which, while perhaps not on equal footing with Gregorian Chant amongst many Roman Catholics and Catholic musicians, is a chant form worthy of particular attention, in light of its appearance in the liturgies in Catholic churches within the Anglican Ordinariate. In addition, it is a well-developed liturgical chant form with a long tradition that adapts well to vernacular English Psalm and other texts, and, as such, is already making inroads within Roman Catholic music.

I'm not sure what other music in the Anglican tradition might fit in an Anglican Music category alongside Anglican Chant. It seems to me that, once one moves outside the realm of chant in the Anglican tradition, one is immersed in the world of sacred polyphony and hymnody (and perhaps some vernacular plainsong), which are already represented by categories here.

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In projected discussions, please do not commit the error of classifying all chant in four (or more; or three) parts as "Anglican." Harmonic chant, elaborating upon the Gregorian tones, emerged during the Middle Ages (so "Gregorian in the tenor" chants are not an Anglican invention); and this is to say nothing of Eastern chant.

To fail to trace back to the origins of Anglican chant to Gregorian chant here would be like disavowing knowledge of a relative who was a known scoundrel...

Perhaps something on the order of
'Harmonised Chant Genres', or
'Anglican Chant and Faux-burdon', or
separate categories, like..
1) Anglican Chant
2) Faux-burden

-As Adam intimates, there IS, really, a difference between 'fa-burden', and 'faux-burden' (and 'falso bordone'). The former is, indeed, an English genre, of XV. century pedigree, in which improvising added pitches to a cantus firmus results in a series of sixth chords. This was also known as 'sight' singing, as it was improvised 'at sight' according to received conventions. Faux-burdon is a French technique, also of at least XV. century pedigree, of notated harmonised chant in which two of four voices were written out. The Spanish falso bordone is not to be confused with the French faux-burdon: in it, all four voices are notated.

Nowadays, the three terms are commonly assumed to be synonymous of a single style or genre and are used rather interchangeably. One often sees any kind of harmonisations of a chant cf called faburdens, fauxburdons, and, ocassionally, falso bordone. Actually, while obvious cousins, they are distinct categories.

How far we have fallen... when we can't sing any part music at all of any kind that is not fully written out by a 'composer'. We would be amazed at the improvised polyphony of which our forbears were masters.

I think that a category called 'Harmonised Chant Genres' would be really good, and would cover them all.