"I think every serious airline company need to have internal
examination system to prevent from flying people with mental problems".
Good idea, but it's not enough.
The authorities & airline managements must have full access to a pilots,
* financial situation, (mortgage, debts, gambling history)
* marital situation, (married, divorced, separated, dependants)
* personal health history and that of all family members
* lifestyle choices, (gay, cross dresser, transgender, hellfire club membership)
* traffic violations
It's our employers right to know everything about their workers to determine their fitness to fly.
Actually, i could live with that. However, all that information, from everyone who has the right to ever enter a flight deck, has to be available to those that control entry to the flightdeck anyway, the pilots. Dunno about other companies, in my outfit everyone on the operational side of the airline will at some point or another enter the flightdeck. So every pilot, flight attendant, maintenance engineer, everyone from documentation, crew planning, control, revision (they do jumpseat flights), operations office, flight planning, network operation center, the whole management, auditors from the authority, partner airlines, IATA, every crew member of any airline worldwide (as those are allowed to jumpseat) and so on.
Sadly, that will make it impossible, so we have to rely on our judgement, as usual.