I would be quite shocked if women who regularly get smears, now stop getting them because of this campaign.
The muddying of the waters WRT the word 'woman' can stop women getting smears. It's becoming increasingly difficult to get an appointment with anyone, let alone a guaranteed, card-carrying cervix haver. I won't go if there is the chance a man will be performing it. And I don't want to risk having to go in and find out when I'm in the room. And refuse and deal with the inevitable accusations of some phobia. Easier to swerve it.
If there was a campaign as mentioned above, with he actual women who do the smears, reassuring us, that would help. Advising women how to request female staff, let them know about trauma, book a double appointment if they struggle, tell them how to eat a support/interpreter/chaperone. And so on. Make it seem warm, kind, female and supportive. Not 'empowering'.