On the contrary, that video actually supports the OP's claim. A pilot with a
well-managed mental condition would be allowed to fly. So, a pilot with depression over the loss of a parent would still be allowed to fly.
But a pilot with suicidal ideations, or taking hallucinogens, like Emerson from Alaska Airlines Flight 2509 (the flight discussed in the linked video), would not be allowed to fly. And that's exactly the way it should be...
The issue is that many pilots don't seek treatment for mental issues because they imagine that they'll be grounded if they do, even though FAA policy, and most airlines' policy, is to allow them to continue flying.