To start with, "visibly insane" is a ludicrous idea.

Further, yes, you can get a license despite being "visibly insane". Some people consider it insane to believe in a deity, to not to, to smoke cigarettes, to have dyed hair, to have an electric car, to have a lime car, to own a Nissan, etc. During my test, the tester said nobody should drive if they hadn't gotten 10 hours of sleep, and I admitted I had only had six or seven. I still got my license.

Interpreted generously, maybe some small percentage of "insane people" are visibly so. Even then, looking at someone is not a proof of a lack of sanity.

Sanity isn't a thing you can test for, it's definitely not something the government should test for.

To make the implied question explicit: How do you envision positively testing for sanity? How would you feel if this test were implemented in 1940, 1900, 1860, 1820, etc? Would your feelings about that differ by the inclination of the politicians in power?

are you aware of the history of mental institutions and insanity charges being used to suppress political dissidents around the world? Most notably in the USSR but the west has also done this sort of thing.