---------------------SPOILERS--------------------------------------- You have to keep in mind that when she was a very small child she witnessed her mother's lover commit at least one brutal murder while her sadomasochist mother was getting off watching it.
She was raised by a woman who achieves sexual release tied up watching girls get hacked, slashed, and strangled to death. That does not make for a healthy home life. I think it's pretty easy to conclude that her mother would have employed all sorts of emotional manipulation and negative reinforcement to ensure that her daughter never snitched on her. It is also likely that at her impressionable age, Betty might have been deeply confused by what she saw. Is this just something that adults do, etc.
Betty obviously looks up to her mother...I mean...she's become an opera singer just like her. If mommy likes it it can't be bad, can it...mommy can't be bad, can she? She couldn't tell the police on her mommy or this mysterious hooded fellow she associates with mommy.
Betty has a lot of deep-seated emotional issues. Her mind has for years been trying to block out the memory of what she saw her mother doing...but it keeps coming to the surface, manifesting itself in the form of horrible nightmares, skull-throbbing migraines, a dependence on relaxation techniques, and sexual frigidity She associates brutal violence/bloody death with sex on a subconscious level. There's an inner struggle between the part of Betty that has confused murder/sex and the part of her which believes these things to be wrong.
After she's seen her boyfriend murdered by the hooded man...she calls the police, yet is unwilling to give her name. The part of her that thinks murder is wrong forces her to make the call, but the part that is ambivalent won't allow her to admit personal involvement. The ambivalent part of her takes control before she can go all the way. So she walks away from the phone in the rain...and when she's picked up by the director she's acting surprisingly calm, not as upset as you would think a "normal" person would be...because the part of her that's been blocking stuff since she was a child is trying its damnedest to block the horror of what she's just witnessed.
The state of affairs in her life all contribute to an impasse within Betty's psyche. Her singing career is starting to bear fruit...she's going to be a great opera singer like her mother was. But is she going to become like her mother in all ways? In the darker ways? Or will she be able to make her own path? Add this to the re-emergence of the hooded man murdering everyone around her.
It's not until the hooded man kills Daria Nicolodi's character that Betty really takes an active role in defeating the killer. Here's someone who loves Betty, who's supported her wholeheartedly in her emerging career, who is in fact a maternal figure in Betty's life now since mommy's dead. Imagine how terrible it would be to lose your real mother and then to see the woman who is the closest thing you have to a mother get shot through the eyeball.
I could go on...but I won't. The main gist of what I'm saying is that the character of Betty is a lot more complex than most of the reviewers on here have been willing to acknowledge.
Opera is one of Argento's best...and not just for the visuals alone (although they are truly magnificent) and not just for the inventive murders (although they are). There is a depth here...and attention needs to be paid.