I think that brightness & edginess are 2 separate qualities that are often lumped together when they should not. For example, Wilson speakers are edgy in my experience as an owner, but not bright. Some Aerial speakers (one owned 10ts) can be bright, but are anything but edgy. I once tried some Nordost Valhalla ICs on my Accuphase 75V, & they were so bright & forward they drove me against the wall, but at the same time they were smooth.
Brightness is more tizziness, glare, forwardness, or screech.
Edginess is a kind of etched, brittle, scraping, hash-like, rough sound especially noticible in vocals and piano notes.
While both are bad, edginess is far more annoying to me, because it seems more intrinsically part of each instrument or voice, making each sound artificial. Brightness be sometimes be listened through, like clicks & pops from vinyl.
The opposite of bright is dark, and I have heard a system sound dark while still being edgy.
Just my 2-cents on this subject.
For what it's worth, in trying to avoid edginess, I like Shunyata Anaconda Alphas in the high-price range, TG 688s or SLVRs at the modest-price end.