Let’s see if I can distill this answer to address the concerns of a somewhat rambling OP.
Pap smears and cervical cancer
Pap smears are currently the most effective means to detect early-stage cervical cancer. It’s relatively inexpensive, quick, easy to do, and cost-effective, and can catch it at an early enough stage to get it all and cure the woman. This has saved a LOT of lives.
You are, however, incorrect regarding the frequency. They should start when the woman becomes sexually active. That means if a 15 year old girl is having sex with her boyfriend(s) she should be getting a pap smear every year and NOT wait until she is 21. That is because sexual activity increases the likelihood of cervical cancer due to sexually transmitted viruses. That means if an 8 year old girl is raped she should get pap smears because her rapist may have given her a virus that could trigger this cancer. Women who are sexually active with multiple partners should get a pap smear every year. The three year interval is only for women who are either celibate or have had a single, long term partner and are otherwise at low risk for cervical cancer. Women past 65 are unlikely to have cervical cancer because either they’ve never had the virus or their systems fought it off, BUT women who are sexually active past 65 with multiple partners might be well advised to continue with the pap smears (yes, I’m talking about Old People Sex. It happens. They’re old, not dead.)
HOWEVER - not all cervical cancer is due to viral infection. Like any other part of the body, cervical cells can malfunction. A WOMAN CAN GET CERVICAL CANCER EVEN IF SHE IS A VIRGIN. Just like non-smokers can get lung cancer. It is rare but not impossible. Hence, even if a woman is a virgin she should get a pap smear at 21 and thereafter because she might be one of the unlucky ones with cells that turn cancerous on their own. It can happen, and since pap smears are easy to do and relatively cheap it’s been deemed a reasonable thing to do.
This also why young people should get the HPV vaccine prior to sexual activity - because it prevents infection with the human pampilloma virus which can cause genital warts and can trigger cervical cancer. It further reduces the chances of people having this.
Also - a pap smear isn’t just blindly ramming a swab up a woman’s hoo-ha. It’s always accompanied by someone looking inside the woman’s hoo-ha, which can also find other problems at an early stage. While it is primarily for detecting cancer other problems might also be found that can be treated. Like, for example, genital warts that would otherwise be unobserved.
CT Scans
Yes, sometimes CT scans come up as a false positive. That’s possible with all medical testing. You seem really focused on the cancer aspect, but really, even benign tumors can cause a problem if they’re large or interfere with other anatomy, ditto for cysts, they can find abscesses that, while non-cancerous, should be drained and treated. They have a role in determining precise anatomy when there’s a deformity or other problem that needs to be addressed, which greatly simplifies the surgeon’s job and reduces time on the operating table and some sorts of complication.
Yes, sometimes it turns up completely harmless lumps and bumps that are then biopsied, which carries a small risk of going wrong. In fact, we’re getting so good at detecting these things that doctors have found them to be much more common than previously supposed. There is a an on-going discussion among medical sorts about how determine what needs further investigation and what doesn’t.
However, a biopsy that comes up “harmless” is not going to lead to unnecessary surgery. Some things are indeterminate or may be causing or about to cause a problem and then whether or not to do surgery does become debatable, but just because something isn’t cancer doesn’t mean it’s harmless or shouldn’t come out. A benign tumor might be CT scanned prior to surgery to better determine exact size, shape, location, location of nearby important stuff like blood vessels or nerves, and so on so the surgeon can better plan to get it all while causing the least amount of damage to other tissue. It might still be biopsied to determine if is a localized cancer, or arising from an infection, or something of the sort.
Before CT scans there was something called “exploratory surgery”. That might have entailed unzipping you from crotch to sternum and mucking about in your guts, looking for what’s wrong or doesn’t look right, then sewing you back up again. It was a LOT more invasive, painful, and likely to cause complications than a CT scan, even a CT scan with a false positive on something because even with the false positive they aren’t randomly digging around inside you, they have a specific target.
Bottom line: not perfect, but better than what we had before.
Mammograms
Remember - “mammogram” is just a fancy word for “x-ray of the breast”. Everything I just said about CT scans apply here. Not perfect, but better than what we had before. we can find stuff that can’t be felt, we can more precisely target things like biopsies instead of simply lopping off bits “just to be safe”, and so on.
However, mammograms, like all x-rays’ are not perfect. You’re basically trying to see through a solid object. If tissue gets dense enough it will block the view. That is why sometimes an x-ray can miss something like a small fracture in a bone - the surrounding bone might hide it unless you are looking at a very specific angle with the x-ray tech may or may not happen upon by accident (that’s why it’s standard to take multiple views of something with an x-ray, multiple views reduce the chances of missing something) Younger women have denser breast tissue, so it’s more likely a mammogram will miss something. Imagine breast tissue as fog. Young women have a very dense fog, the sort where you might not be able to see a building across the street. Older women have less dense fog, where you can see farther and, sure, that building across the street might be a bit fuzzy around the edges but now you can see it.
Sure, you could increase the intensity of the x-rays for a better view… but increase them enough you could, potentially, cause a burn and/or increase the chance of cancer later in life, so there’s a cost/benefit consideration at work.
Do we need to change how we do any of this?
A perfectly valid question. Yes, sometimes we do need to change how we do things. At one time pap smears were new and controversial. Then we said they needed to be done every year without fail. Then we learned more about risk factors for cervical cancer and could divide women into “high risk” and “low risk” groups and say the high risk folks needed them every year and the low risk folks every two or three years So yes, we do change things over time.
Likewise with mammograms - when to start getting them, who gets them, how often, and so forth are all questions that are debated within the medical community. Same with CT scans. Remember, too much x-ray can cause damage, too, so the need to screen people and know more has to be balanced against possible damage from the testing. Modern x-rays expose people to less radiation that those from 50 years ago, but while each “snapshot” of a CT scan exposes you to less than an x-ray from a couple generations ago a CT scan is composed of hundreds of such snapshots and therefore adds up to more radiation overall. There have been people injured from getting too many CT scans.
So… who gets tested for what in what manner and how often does change over time as our knowledge grows and our testing techniques get better. but NO medical test is perfect. That’s why, when something comes up “positive”, there is further testing, often with different techniques, in order to catch those false positives. False negatives - when something that is bad is missed because no test is perfect - is part of what sets the testing intervals. You want to test often enough so if you get a false-negative for whatever reason the next test will catch the problem soon enough to do something about it, but not so often the testing itself starts to cause a lot of problems.