If I can step in with a personal anecdote: I was diagnosed in my late 20s with papillary thyroid cancer.
PTC is a very survivable cancer, with a near-100% survival rate (death usually only occurs in rare cases where the disease is diagnosed very late, the progession is atypical, or there are comorbidities at play). It is very easy to screen for and diagnose: a neck ultrasound identifies thyroid nodules, and if the nodules look suspicous they are biopsied in a 20 minute procedure performed under local anaesthetic.
Treating papillary thyroid cancer is also relatively straightforward, as far as cancer goes: depending on the size of the lesion and the features, either half or all of the thyroid gland is removed surgically. In cases where the whole gland is removed (which is the majority), the patient is given a course or two of radioiodine therapy to nuke anything left over, and in many/most cases, it's a done deal.
The vast majority of thyroid cancer survivors have to take thyroid replacement hormones (all patients who had the whole gland removed have to do this, and about half of patients who only had half the gland removed still need a small dose to keep up). I'm relatively lucky: the oral hormone seems to work just fine for me. I take a pill every morning and then go about my day. I will need to do this for the rest of my life, but hey, that's life.
However, there's a substantial minority of patients who aren't so lucky: even with oral hormone replacement, they suffer from long-term sequelae including weight gain, low energy, brain fog, hair loss, and other hypothyroid symptoms.
And there lies the crux of the issue: it turns out that even with increased diagnostic capability (thanks to the ubiquity of relatively cheap ultrasound exams in clinical practice), the number of people dying from thyroid cancer has stayed pretty much flat for decades (mostly due to more aggressive types than papillary, such as medullary or anaplastic). Yet, we take out a lot more thyroids now.
The reason this happens is pretty simple: if you see something, you have to do something about it. So you're removing thyroid glands from people where the cancer might never have actually grown big enough to be a problem, and then subjecting those people to a lifetime of hormone replacement therapy. Something like 10% of all cadavers at autopsy have thyroid cancer: it's a cancer that very commonly develops, but only becomes a concern in a few patients. As of now, we don't have a good way to differentiate between "thyroid cancer that's a problem" and "thyroid cancer that'll be fine."
The clinical guidelines have changed a bit in recent years: if the cancerous nodule is really small, they'll now do "watchful waiting" and monitor the nodule to see if it grows. But you're still subjecting a patient to potentially many years of worry and regular testing. And good luck getting life insurance if you have a microcarcinoma! Yeah, it's highly unlikely to kill you (especially when monitored), but try telling an insurer that.
The medical profession is well aware of these concerns. That's why they avoid testing for thyroid cancer unless there are symptoms, such as thyroid hormone disturbances or a lump in the neck. If you were to make a thyroid ultrasound a regular test, you'd quickly overwhelm the system with cancer patients who probably never needed to be treated in the first place, and who may now have to get their thyroids removed and be dependent on pills for the rest of their lives.