I would love this idea. I always enjoyed it in DA2 and DAI when characters (including my own) recognised that I was a mage (or my race, specialisation etc). The thing with the engineer in Omega was pretty good. I was always disappointed when you'd get things like Shep forgetting to use their biotics when it would give them an edge over non-biotic Shep. Or Gabby saying "you're boring the commander with tech" about my engineer Shep.
I can't help but agree with bob/david/babypuncher a little about getting an edge in cutscenes because of class. While there is the scene in Omega, it ultimately made no difference since you can save everyone regardless. While I like having class and abilities flavoring the scenes, I don't think that that story sequences should have totally different outcomes gated behind class. To some extent, Inquisition has something like this, insofar that certain quests cannot be completed without a certain class, but this doesn't depend entirely on the Inquisitor's abilities, since you'll always have one of each class to chose from no matter what to comprise the rest of your group.
But biotics or tech should never, in my opinion, drastically change the big plot points. I greatly favor the Vanguard over all others, but I don't think that being a Vanguard should suddenly give me an auto-win in a scene whereas a soldier or infiltrator Shep gets nothing. With six classes to choose from, it gets rather complicated, and ensures that someone gets left in the cold.
Now, I do support the different classes being reflected somewhat in cutscenes to at least flavor the sequences. I rather enjoyed the way the Citadel DLC reflected this when dealing with the clone.
As for dialogue, there is one bit that I recall that reflects whether or not Shepard is a biotic. When Traynor mentions Grissom Academy, she says that my Shepard would probably have gone there if it was around longer, and then there's Liara's dialogue during the capsule scene, which even differentiates between the types of biotics, since she mentions my Vanguard charging into a fight. With the Donnelly dialogue you mentioned, I always thought of this as an example of brevity being more valuable than tons of jargon. They all have jobs to do, but dealing with the intricate workings of the Normandy's power systems is not the Commander's. The long-winded explanations don't mean much. He just needs to get to the point.