1. Lending your credibility when boosting from outside your circles.
If I boost a post by someone none of my followers know, they're unlikely to re-boost because they haven't vetted the source. If it's quoted with my comments, though, anyone who knows me can boost and delegate that vetting to me. This makes it possible to introduce outside voices and build networks across disjoint communities.
2. Adding content warnings.
I can't expect others to put CW on everything that may be triggering/day-ruining to my followers. If I want to boost an important story that might be, I want to quote it and put that behind a CW, so my followers can decide whether to expand. I don't want to boost it unless it's already under CW.
5. Re-upping your own old posts/threads/art/etc.
Yes this can be done with normal boost, but then you end up with nobody realizing it's old and everyone in the replies to the original suddenly getting @'d over and over. Doing it as a quote reduces this effect (empirically! as observed on the other site) and makes a new context for new discussion.
@dalias It seems to me that all of these usecases would still work if they had cc semantics similar to replies or to boosts (i.e. by default/always cc the author of the quoted message/the cc list of the quoted message).
(Note that cc lists for replies are already weird: it's easy to accidentally send a reply that isn't actually sent to the author of the message you're replying to using some clients. Mastodon-the-service considers such replies to be valid enough to accept from users and federate outward.)