Your fears of eye damage are completely justified. But with apologies, I have to note that you are mistaken on some key points.
And these are the same mistakes that Tyler Nordgren (the astronomer quoted above) and I saw happen in Oregon in 1979.
The guidebook for many observers of the 1979 eclipse was Bryan Brewer's Eclipse (the link is to the 2017 edition). Sadly, Brewer notes that many people ruined their vision by improperly observing a previous eclipse:
> During the March 7, 1970, eclipse in the United States there were 145 reported cases of people who damaged their eyes by looking at the partially eclipsed Sun either directly or through sunglasses, exposed film, smoked glass, and the like. None of these homemade devices can be guaranteed safe. Instead, use professionally manufactured “eclipse glasses” made with optical grade film that filters out most of the Sun’s rays to protect your eyes from damage during the partial phases of the eclipse. Play it smart and don’t take any chances with your precious gift of vision. And… don’t forget to remove your eclipse glasses during the minute or two of totality. Otherwise you’ll miss the beauty of the solar corona, which is not visible through the filter material.
For young children, even with eclipse glasses you would be right to worry that they may not use them properly, and may even try to sneak a peek at the partial eclipse without them. But there is another method that is entirely child-safe. Brewer continues:
> In addition, there are some perfectly safe ways to observe the partial phases of the eclipse without looking directly at the Sun. These methods involve viewing the image of the Sun projected onto some surface; the image can be focused by having the sunlight pass through a pinhole… This is a safe and recommended way to observe the passage of the Moon across the face of the Sun during the partial phases of a solar eclipse… And don't forget: during the few minutes of totality it’s OK to look directly at the Sun's corona.
The KidsEclipse website has instructions for making a box pinhole projector like the one described in Eclipse, along with a list of safety facts. NASA's Jet Propulsion Laboratory has instructions for a simpler version. Or you can just use two sheets of paper: poke a pinhole through one and let the sun shine through the hole onto the other.
With a pinhole projector, you don't need to be "exceedingly careful." It is no more dangerous than being outside on any sunny day, because you are not looking at the sun at all during the partial eclipse. And it's easy for an adult to verify that a child is using the projector safely: you simply make sure they are looking away from the sun, not toward it.
You mentioned clock accuracy, but that doesn't come into play at all. You don't use a clock to tell when totality occurs. The sun itself tells you; it's like going from day to night in an instant. And again, directly viewing the sun's corona during totality is completely safe. You can even use binoculars, as I did in 1979.
There are many activities that can be dangerous for children but beneficial when done right. Just as you would never say, "jump into that river and teach yourself how to swim," it would be foolish to let a child go out and watch an eclipse on their own without preparation and adult supervision. And it is tragic that 145 people damaged their vision by not using proper viewing tools during the 1970 partial eclipse.
But with the very simple precautions that Brewer, KidsEclipse, and NASA describe, anyone can view an eclipse with absolutely no risk of eye damage.
No one who views a total solar eclipse will ever forget it. Nothing on this planet compares. It is wondrous and awesome. You experience the glory of the Universe.
And you can do it in perfect safety, children and adults alike.
It was truly a shame, a gross dereliction of the educational duty of the Oregon schools, that so many children in 1979 were denied the opportunity of a lifetime. They were not allowed the awesome experience of totality, because adults chose to be governed by fear instead of educating themselves and their children on how to make the experience perfectly safe.