I recall a similar discussion when I was trying to sort out the Goldberg Variations. A useful contribution in that case was
In other words, while important, the catalogue is only one consideration.
Unfortunately, the style guidelines don’t seem to say much about naming works (as opposed to track titles), but the general approach seems to be:
Generic name & number, catalogue number, non-generic name
Reviewing the entries for Bach, it seems to me that those which have received most attention have the structure:
Generic name, Catalogue number, non-generic name
(I.e. omitting the generic number even if there is one)
Also, the generic name is in German not English (Kantate not Cantata).
The source you quote is interesting, but is one of many (Wikipedia, for example) and they are all slightly different. Also it seems to be English-biased (e.g. “Christmas Oratorio”).
So my view would be not to remove the generic names, but to make them consistent and German.
EDIT: given the potential widespread impact of this issue, a comment by @reosarevok would be helpful.