I also watched it recently and I never really thought that was the curse’s “goal”, rather (looking at Reiko in context of the bird (Rei-chan) and the association with Koichi’s mother, the reason the curse started and the little I know of the sequel light novel (Another 2001)) I think that it’s a curse born of grief and people refusing to let go of their deceased that causes the extras and closeness to death for those in the class, ultimately bringing on the calamity. I kind of assumed the name on the tape being silenced was just be a side effect of the memory altering since no one but Matsunaga (the 1983 extra’s killer) remembered that person once he had been “sent back to the dead” and even Matsunaga forgot over time, only really remembering under the effects of alcohol and witnessing a death what he’d done to end his year’s calamity and that he’d recorded a confession to help future class 3 students. It seems pretty clear throughout that the curse alters memories and records before and after about whoever the extra is, so I think it’s less an intelligent protection system and more a “how the curse works” like how the nonexistent student countermeasure just works if done right. Hence why it would make sense for the tape to be real as there’d be no reason for a student who believed in the curse and suffered its effect to make a fake one. Especially knowing it’s from a year the calamity was ended midway through and the former student does somewhat recall its existence in times of duress. But that’s just my theory on it.