The Eye of Insanity cannot trigger that endgame condition.
You are required to choose the ability at the beginning of your turn, but actually using it instead of the crew’s original ability is optional (the text says “can”, rather than “must”), so the crew’s original ability is still considered active/available when that endgame condition is checked.
The Eye’s effect only lasts until the end of the player’s own turn. It does not carry over to opposing player turns, and then has to be reset at the beginning of the player’s next turn, even if they do choose the same ability again.
Side Note: The Eye can prevent that endgame trigger as long there’s at least one eligible ability somewhere in play that would allow the ship to still move, repair, or get to a repairable location in some other way.