The FFX one is pretty iconic. I think everyone that's played the game is painfully aware the scene is intentionally awkward and bad sounding, but intentionally unpleasant is still unpleasant at the end of the day. Especially if the gap between the normal performances and the ironically bad performances is really not all that big. Still, I think we've got more worthy examples of awkward scenes that don't devolve into thematically appropriate, awkward discussions about people "not getting it". Looks like Star Ocean 4 has been covered already, and it's the crown jewel of this shit. The game itself itself is mostly fine, but holy hell the cutscenes test your extremities of your patience.
I should note though that there is a type of flavour of badness that I do enjoy quite a lot. Devil May Cry's "I should have been the one who filled your dark soul with liiiiiiiiiight" and Mega Man X4's "What am I fighting fooooooooooor?" are pretty bad, but I think they have a certain passion and earnestness to them that I find really endearing.