When angels take on a full physical human appearance to do an angelic rescue of a person in danger, is it truly necessary for them to have full physical form and density to carry out their mission, or is that done to not force the witnessing of a miracle on people who are non-believers?

You have asked and answered this quite nicely, as you are in tune, and saw the circumstances intuitively, and factors governing such phenomena. An angelic rescue is serious business, for serious purpose, and must be done in a way to appear seamless and natural, and for the reasons you cite. It is not always appropriate for the person rescued to be aware it is of a divine source, and that is for many reasons, from the level of their belief, to the nature of the karmic circumstances involved in their dilemma necessitating the rescue, and so on. So this is a judgment call of divine realm to decide when, where, and how, a rescue can be offered. There are many times when people are in dire circumstances and may not be much of a believer, but it is not their time to transition. And in those circumstances, a divine rescue is warranted to keep their life going, and allow them extra innings, to give them the chance to be awakened or complete planned tasks they may be unaware of that help others or themselves, that, may be quite important but not necessarily dependent on the conscious awareness of the person, as to their necessity and the involvement with their karmic destiny, and so on. So there are many compelling reasons to keep a life going once it is launched, and this will be done in many circumstances when the person is not truly at an exit point that is preplanned.