I really hate mammograms, having large dense breasts that are very firmly attached to me. Its always a struggle for both me and the radiographer, and her empathy and kindness is very important. I would want a woman.
I can see the point about vacancies, but I think this is a systemic issue around pay, conditions and retention, and you would be robbing Peter to pay Paul, just moving your radiographers around. It would not solve the problem. The NHS is short of qualified staff in every area.
If we did have enough radiographers and a choice was available, then ok, but it must be clear and upfront. People don't always think about how this would affect religious and the elderly too. A close relative lied about a serious condition rather than let a male doctor touch her in a private place, and it killed her.
The evidence around breast screening is a bit murky, there being a lot of false positives, so if there was no choice I would probably just opt out.
I came away from my first mammogram distraught at how painful, undignified it was, and how little I was informed about this (though comms are better these days). Similarly with hysteroscopy, and intravaginal ultrasound etc, which were both unpleasant, undignified and painful. I think women's pain, dignity and discomfort is minimised, especially if its not all women affected in the same way. ( 'Well it doesn't hurt ME')
What I will suggest is that if any of the screening tests for male cancers were as painful and invasive there would be a an alternative such as a blood test by now. Like the PSA.
What is actually needed more money in the NHS overall (not more reorganisations), and more research into women's health.