I apologize if what I am about to say comes of as inconsiderate and I am by no means attempting to be rude. Sometimes, it is best to call a spade a spade.
You'd be amazed at how many things optical engineers from all sorts of companies will politely nod through in order to avoid pissing off a gunwriter or a customer, or simply to avoid getting themselves into a protracted explanation of something to a lay person who is not going to put in the effort to understand any of it. You are not the first gunwriter (or a customer within industry) who thinks asking questions from people who have a vested interest in flattering you is productive. Nor the last. I've been on both sides of this phenomenon.
My favourite example of that is probably another gun writer who set-up a blatantly stupid image quality test in his backyard. Then he asked some marketing folks from a couple of optical companies whether that is a good test. Of course, they said it is. If they said no, they either risk offending a guy who clearly think he knows what he is doing or they risk getting themselves into a lengthy tutorial on how to set-up a proper scope test. The gunwriter in question, while generally a rather reasonable guy, can't tell his ass from an elbow as far as optics go. However, after twenty years of scope manufacturers flattering his "brilliance", he is convinced he understands optics.
I am sure you are more circumspect than that, but you see my point. If you want a clear and detailed answer, ask someone who does not care about hurting your feelings. Sitting through meetings with optical engineers without doing some extensive homework on the subject will generally qualify you to bullshit convincingly to an audience that knows less than you do. Until someone in the audience does some homework. Then, instead of being convincing bullshit, it is just bullshit.
Now, onto some specifics, I think you misunderstand how a human eye perceives brightness. A lot of it comes from image fidelity. Equating illuminance with perceived image brightness is terribly misleading and generally wrong. It does not do anything to illustrate your point on objective size. I reiterate: PERCEIVED image brightness.
Perhaps, I do not understand what you are trying to prove. A 56mm objective at the same magnification, will get you a larger exit pupil than a 50mm objective. Given similar image fidelity, a 56mm scope will look brighter at magnifications high enough to keep exit pupil, rather than the eye pupil as a limiting factor. If you are trying to make an argument that XRS-II will have better image quality than competing 56mm designs because it was easier to build, you have to be specific which competing design it is better than. I do not think you can make a blanket statement like that. If there are similarly priced scopes that deliver similar image quality with a larger objective lens, Bushnell will lose some sales to them.
Building a scope is a compromise. If Bushnell wanted to use a larger objective without too many additional complications, they would have to make the scope a little longer and a little heavier. They chose to go with a smaller objective and a smaller overall scope. That is a perfectly reasonable choice. Beyond that, not having seen XRS II, I really can't comment.
When the original 3.5-21x50 Bushnell HDMR came out, they made it too short. Because of that, there was a lot of sample-to sample variation, sicne it really looked like they struggled to keep the tolerances. The one I owned was superb. Others I have seen varied. I did not test the original XRS thoroughly because when I looked at it briefly, it was apparent the optical system there could not quite maintain image quality all the way to 30x, so I passed on it. I am sure XRS II will be better.
Personally, I valued light weight and compactness, so I am not terribly obsesses with objective lens sizes. However, I also do not shoot at longish ranges in the middle of the night a whole lot. If I did, you bet that at least one of my rifles would have the rather excellent 6-24x72 Hensoldt. At 24x, this is far and away the best low light scope on the market (outside of intensified or thermal sights).
ILya
Sorry for the slow reply but it's been a busy weekend.
Gun writers are an amusing bunch for sure. Much like the online community, they run the gamut from super good dudes that are highly knowledgeable to semi-retarded drama queens that no one likes to be around. I'm not sure where I fall in that spectrum, but I try not to worry too much about the mistakes others might be making and focus on trying to make as few of my own as possible. In this thread I have made none.
Of the handful of concepts covered in this thread, how the human eye works is by far the easiest the understand. As you pointed out earlier with what role f-number plays in the optical system, image fidelity absolutely plays a major role in perceived brightness. A bright blurry image is useless. Smaller objective lenses more easily produce good image fidelity than larger lenses because, well, f-number is focal length divided by lens diameter and math is an unforgiving mistress.
I'm not trying to prove anything in this thread, nor do I have much desire to argue. Large objective lenses are very misunderstood in the riflescope consumer market and I'm trying to help those who frequent this forum learn how to develop well-informed opinions of their own. Confusion benefits no one except the marketing department. I believe the layman can learn enough about how riflescopes work to eventually eat the optics elephant. He just needs to know how to do that one bite at a time. Riflescopes are not an unknowable mystery.