Not all woman have a cervix and not all people with a cervix are women.
I don't really see any problem with the wording of the campaign.
I regard myself as an intelligent woman. I'd had a hysterectomy and a few years later queried with my GP why I hadn't been called forward for the regular smear test. I felt a tad foolish when I was told I no longer had a cervix. I had no idea that it had been removed during the operation, had no idea that there were two different kinds of hysterectomy - total hysterectomy where the uterus and cervix are removed and and supracervical hysterectomy where the cervix is left in tact.
Incidentally, it's a common misconception that a total hysterectomy includes removal of the ovaries. If the ovaries are removed, that's a bilateral salpingoophorectomy or BSO.