>The disease [...] are, in the general population, pretty rare.
Cancer is not rare, particularly from a certain age. Early detection heavily skews the odds on one's side. This is such common knowledge that it feels weird having to justify it here.
I know I'm not a "survey performed under the most strict standards of quality" but among my close groups of friends and family there have been a few cases where they got lucky enough to detect this disease very early on, and their treatment and outcome was significantly better (and cheaper, while we are at it, bc you mention cost as well) than if they've just waited even a couple more months.
OTOH, I have yet to hear a story from someone who had some sort of scan done, found something worth to take a look at, that then turned out to be nothing and, for some strange reason, had its life significantly affected by that.
Doctors are not stupid, they know (or intuitively correct for) this Bayesian Base Rate you mention, plus they have other things like years of experience in their particular field. It's not like someone's going to cut your arm the moment you find a weird looking lump growing inside.
>these scares can absolutely fuck up a month of your life
That sounds quite immature, tbh. It's definitely a small price to pay compared to all the trouble you save by potentially detecting issues early on. "Oh, but needles hurt :'(", sure, but their benefit far outweigh such small nuisance. This is also an argument I would've thought anyone would just get naturally, and yet I have to explicitly bring it up here, weird.
Also, if it came to be the case that everyone got scanned regularly, that massive trove of new data would definitely help improve the accuracy of said tests. It definitely won't make matters worse, but better.