Now I'm confused. Maybe I don't know how quotes are supposed to work in Mastodon.
When I tap the Quote icon in Fedilab, the toot I was viewing at the time appears above where I can put in my own text. But when I send, no trace of the "quoted" toot appears, and my toot isn't even linked to the quoted one. So Quote just acts like a limited Reply.
Oh, well, sorry for the confusion.