That's correct, the links are one-way from group to story. Stories can be added to groups without the involvement of the author so it wouldn't really be proper for them to appear on the story page itself, the author might not appreciate it if the StoriesThatSuck group added it or whatever (there actually was a problem with a group along those lines on FiMFiction for a while until it got shut down by the administrators).
I was also trying to make sure this suggestion was useful in contexts other than just FiMFiction, too, and I figured it was hard to get more generic than "here's a list of fanfic links, apply this category to them when you download/update them."
Anyway, having a temporary mark applied to the most recent batch of downloads would be a good middle-ground option, I think. It's an extra step but it would allow me to add group tags to a completely different special column devoted entirely to that rather than mixing it in with the generic tags, which would make the tagging safe from harm should I ever accidentally or intentionally take off the "keep existing tags" option.