This is to say nothing about various gynecological exams. "Let me scrape your cervix with a metal tool" is a pretty common procedure, for example.
Also.. your reply informs me that you have no idea what a mammogram procedure involves. And that there are other non invasive methods that insurance won’t cover. And that for most women, the first screening is almost always inconclusive leading to a second one. Multiple unnecessary painful screenings because it’s ‘cheaper’ but also a standard procedure.
When my wife had her IUD inserted, it was the most painful thing she had ever experienced by a long shot, more painful than multiple invasive surgeries and debilitating back pain. The doctors told her that it would feel like "a pinch" and refused to prescribe pain medicine.
I don't think this is comparable.
Digital rectal examination [..] was a common and "dreaded" component of annual exams for older men, as it was thought to be a reliable screening test for prostate cancer. - https://en.wikipedia.org/wiki/Rectal_examination