tl;dw: when a particular engine design used by the 737 MAX (but also other Boeing and Airbus planes) ingests a bird, if there's enough damage, it starts burning oil before the cabin bleed air intake. The cockpit and cabin have air supplied from different engines. Since the cockpit is relatively small, if the cockpit engine was damaged, smoke would fill the cockpit quickly, reducing the pilots' visibility and requiring them to don air masks. Bird strikes only happen at takeoff and landing--times when pilots don't have time to be fiddling with masks and seeing through smoke and trying to shut off the damaged engine. Regulators in the US and EU don't seem especially concerned.
It’s not actually burning the oil, just dumping it into the airstream in the form of tiny droplets. Some of that gets sucked up by the bleed air system, the rest continues on to the combustion chambers to be burned.
Doesn't help you if the pilot flying is incapacitated by the toxic air in less than a minute, or crash in to mountains, as in the scenario posited in the video. Climbing out to an appropriate height to join the approach, reconfiguring the plane for landing etc, takes several minutes; an immediate forced 180 teardrop back to land is very challenging to execute successfully (see the tragedy at Jeju just before new year).
They don’t have to be incapacitated, simply having to deal with reduced cockpit visibility and/or minor impairment during an emergency while in a critical phase of flight is a significant elevation of risk.
