Yes it's not trivial to balance these things and this is why there have been randomized control trials that look at long term end points where both the costs of false negatives and false positives come into play.

It's also why breast cancer screening is not recommended for everyone, just women over 50.

For women over 50 there has been a decades long sincere effort by the medical community to take this all into account and the consensus is that screening makes sense in that population.