A main objection to mammography is the fact that it uses ionizing radiation to take images of your breasts, and it’s a well-established fact that ionizing radiation can cause cancer.
So the idea that the “best” way for you to avoid dying from cancer is to expose yourself to cancer-promoting radiation at regular intervals for decades on end (in order to catch the cancer early) really falls short on logic — especially since there are non-ionizing radiation imaging techniques available.