To my way of thinking, boiling Civ down to a bunch of flippant "always" and "nevers" is ham-handed. Some games don't jibe with the playbook, which is why strategy and tactics are not synonymous terms. Gold is useful, and less fluid in some games than others. Not every civ or map is best-suited to prioritize a hub or harbor. If I'm playing a religious game stuck between Arabia and Russia, then I'd better get holy sites out there or I'm not going to have a religious game for long. Any trade route may have to pass to or through them, and so getting embroiled in an emergency may be pointless. If I'm not already planning to attack them, then the sudden opportunity provided by an emergency may be unwelcome. And if I am, I won't spare myself any grievances for captures, which seemed to be central to your initial point that emergencies present some useful opportunity.
That's just one of many instances where an emergency is worth voting against, "big deal" or not.