One thing that will cause mammograms to be political, in several countries, is that there's a difference in perception of the downsides vs upsides of such screening programmes, and we're bad at communicating the trade to a population who lack the statistical literacy to understand it intuitively.
So our best shot from a public health perspective is to say "Here's what we recommend for everybody" and pay for that.
All screening programmes have two difficulties, which must be balanced against the benefit, and this trade is somewhat personal, so when the balance is quite fine the arguments can be vociferous as a result.
1. The screening itself may seem unpleasant. One woman may find it a very mild annoyance, a drive ten minutes out of her way, the staff are very pleasant, the scan itself is far less traumatic than a bra fitting, and she receives easy to understand results after not very long and isn't anxious about them; but for another maybe it's an hour's bus journey to the city hospital, the staff there are short-tempered and say she has the wrong paperwork, then another hour in a queue, she feels like she's just meat, squashed around for the convenience of the machine for what seems like forever, and then after anxiously waiting for what seems to be too long the results are confusing to her and she has to have a friend interpret them.
2. Over-treatment is always a problem. Screening by definition detects something that isn't causing noticeable symptoms. If you have a noticeable lump, or mysterious bleeding, you don't need screening you need a doctor's appointment. So a positive screening result might be nothing important. However either you've now got the burden of a diagnosis you ignored or, you accept the medical advice and are treated, even though it's possible (not likely, but possible) that you would have been just fine without treatment.
So, screening programmes are set up based on guessing how to trade these factors plus a third, how much should we spend on this medical intervention? After all, in some sense every dollar doing breast cancer screening is a dollar you don't have to cure blindness in poor orphans (or of course, to bomb somewhere)
If your experience of a screening programme is that it's a minor inconvenience at most, and yet you know people who died of undetected disease, more screening seems like a no brainer. Particularly if you live somewhere where screening stops at age 50, and somebody you know died of undetected disease aged 54, you might reason that the screening should go to age 55 or 60 to detect such cases, no matter the public cost.
On the other hand if your experience is that it's an awful ordeal even when negative, and you know people who spent their last years horribly scarred by surgery as a result of suspected disease but then they died in their sleep from something else anyway, you may feel that there's already too much screening and it should be trimmed back, not to save money but the extra money for other programmes is welcome.