One of the criteria for screening tests is that it is an 'acceptable' and 'easy to perform' test. For many women, cervical screening is neither.
Women are good at being guilted into painful procedures, for the good of their family. You can see people doing this with smears- 'you couldn't leave your children without a mother!'
I think the fact that young women appear to be liberated enough that they're saying 'Hey! This test is NOT acceptable to me!' is a positive thing for feminism.
Instead of the patriarchy sitting there saying 'why won't these women do as they're told, like the used to???!' They should be listening to women, and finding an acceptable test- self testing, or blood screening. A self test would be possible when the cervix is nice and low just before your period, and would have the added benefit of teaching women about their anatomy. Allowing a self test, with a teaching appointment, or a nurse to do it, would mean no more appointments than the current system, and I wonder if any more failures than the current system, if you count the number of women who are dragged back to their surgery several times because their cervix is difficult to find.
A screen for men requiring an invasive test into the urethra, or even a routine rectal speculum, would not be deemed acceptable.
The fact that such an invasive test is deemed 'acceptable' is very much a feminist issue, as it appears men decided it was acceptable way back when, and women have done as they were told.
Not to mention the lack of accurate information etc on whether women who have never have HPV, had the vaccine, etc, actually have any benefits that outweigh the risks of unnecessary treatment.