Get-there-itis is absolutely a thing. No big airline would be dumb enough to say it that stuff explicitly but the pressures still exist.
There have been accidents around the world where pilots were trying to save fuel, get off work at a normal time instead of 2 hours extra holding, not wanting to divert and not be home, try to get on the ground so they don’t go over their max hours for the day, etc.
Yes it’s counterproductive. *It happens*. External or internal, it happens.
Get-there-itis is - quite famously - a pilot's internal desire (note: written before your ninja edit).
Sure there have been plenty of incidents. Can you point to one in the US, with pilots flying big planes, that hasn't led to regulatory change?
Its worth pointing out that pilots are almost always paid by flight hour. In other words, they get paid more to go around.