Apparently you're right. I just added it back and I could hear an audible hum which goes away when I remove it. So I guess I will be going groundless for the time being.
You changed
only a twin RCA cable ✅
You're using the same amp as before ✅
Connecting back the originally connected ground cable gave you a hum?
See the thing is, unless someone can point out something I'm missing, when you now re- connect the separate ground cable, you have no more (or less) grounds than you previously had.
In terms of
all electrical connections, nothing has changed between what you had before and what you have now (someone please correct me).
It can only be the case that with the old RCA cable, the same no hum / hum situation occurs when connecting / disconnecting the ground wire.
You may not care (and I wouldn't blame you) in which case, good luck to you
(But I couldn't stand the mystery myself, I'd have to investigate and confirm what's going on 🙃).