The examples you gave are regulated.  Still, there's a lot of dangerous
machinery that aren't.

The most obvious reason why lens prescriptions are legally required is to
minimize the risk of operating dangerous machinery, such as automobiles.
Can you imagine drivers, pilots, ship captains, etc. self-prescribing?
People cannot be trusted to be responsible and use common sense in such
situations.  How could you even tell if you hit on the correct prescription?

I can see why prescriptions are required for drugs, because those can
be easily abused or sold in the black market. But to me, eye
prescriptions don't seem that critical.

My analogy is shoes. You could probably mess up your
feet/knees/legs/back by wearing the wrong size or type of shoes, or
improper inserts long term. Yet a shoe prescription from a podiatrist
is not required for people to buy shoes.

Of course, everyone should get an initial refraction done
professionally. But what's wrong with buying disposable contacts that
are a little higher or lower than your prescription to see how they
perform? You're not going to be seriously injured. Maybe get some
headaches or eyestrain, that's all. And if you do have problems, then
go see the optometrist (just like you'd see a podiatrist if shoes were
giving you problems).