In May 1970, Gary Alinder and fellow members of the activist group the Berkeley Gay Liberation Front attended — or, as Alinder later recounted, “invaded” — the National Convention of the American Psychiatric Association in San Francisco. The activists’ purpose was clear: They planned to confront the more than 10,000 psychiatrists present about the APA’s designation of homosexuality as a mental illness. A new documentary, “Cured,” that premiered Monday on PBS — for National Coming Out Day — chronicles this years-long campaign, which ultimately led the APA to remove homosexuality from its manual of mental illnesses in 1973.