The eye test is a good idea. I believe in Sweden a shooting test is required before a hunting license is issued. I know it controversial, but I have always thought that some kind of minimal shoooting test should be required.
Licenses to operate cars and airplanes require actually demonstrating an ability to operate those things. A surgeon must actually operate on a live person before they are permitted to practice. A teacher must present a lesson to a classroom before they are credentialed. Many skilled endeavors require some sort of test to determine qualifications. Hunters can get licenses without having any knowledge about the proper and safe usage of their guns.
I know many states require hunter safety classes. But I know that in California, the couse does not require actual firearm shooting. You can learn the safe way to oerate a car with a book. But until you are actually driving one all the book knowledge doesn't amount to a hill of beans.