I take issue with your premise as to why pilots do not wish to say the word emergency. I’m not saying you can’t declare an emergency when you see something. I’ve used controllers to visually check my plane a few times and it is a welcomed set of additional eyes. I’m saying, over the decades, I’ve never heard of any controller declaring an in-flight emergency for an aircraft. In that case you have no moral obligation to take the situation out of the pilots (who is dealing directly with the situation), hands.
from my point of personal reference, military and gen av will declare an emergency without hesitation. air carriers will delay (when able) in an attempt to resolve and are further cognizant of the fact that the broadcast of emergency is over open air. and yes, from my personal point of reference the company has a lot to do with it.