I like your introductory paragraph. You should expand on that elaboration, maybe in simpler terms. Beware that I'm from a scientific background. Now your two questions.
If one neither affirms nor denies the reality of ontological necessity, then what justifies positing it as an ungrounded principle in our philosophies?
Who is we and what are our philosophies? If you say that necessity has such an important foundational status, then you're implying we make our arguments depend on it. I don't quite see this. I feel philosophy now is more asking questions than giving answers. I think the only philosophical deductions which are for the ages are of the form "If we agree on these formal chains of reason in analytical logic, then this letters on my page lead to these". It only gets dangerous if one claims that the formal expression "P => Q" can actually be interpreted as saying something about reality - and I think many philosophers themselves think in the modern scientific way you describe: "A formal logical deduction might be used to say something about the real world, but when it happens not to be applicable, then it just isn't and we shouldn't a priori expect it". What I say here is that, just like scientists, many philosophers don't even believe in "reality of ontological necessity", because they are themselves brought up in a more or less scientifically literate world.
Given the pragmatic and hypothetical tone of the sciences, may its reliance on necessity help explain the difficulty philosophy has in being heard and taken seriously as a cultural form?
By cultural form do you mean everyday culture or the academic realm? Either way, my answer is no, I don't think that this is the reason, simply because I don't think enough people think about it to that extent. Time after time, fields of study have been exported from philosophy to stand-alone sciences. When non-philosophers think of "philosophy", they now equate it with "history of western philosophy". And they might be right to some extent; I feel that philosophers often try to philosophize on topic within a field - but to do so, just to grasp what philosophical solutions are needed in that area, they would have to study the field, becoming scientists themselves. If this route is to be avoided, then philosophers "should" focus on the meta questions of the field of study, e.g. why study it at all, what should be the goal etc.
It would be tremendously interesting to me what actually motivates people to argue for this and that. Example: Yesterday I was browsing here on Philosophy.SE and I learned there is a "movement" called Speculative realism. The topic itself has nothing to do with the point but I want to say that there are >9000 point of views, and more you could potentially come up with in the future, which one could defend. Now in the sciences, e.g. if you investigate the brain, or if you work in the field of superconductors, then this is your job and so you do what you do. There is an empirical goal, may it be justified or not. On the other side, I have no insight into what extent people who talk about this and that worldview are actually motivated by some personal desire to... I don't know. Why do they even want to convince someone else? "Rationally speaking", the chance that their point of view is the "right" one (if there is such a thing) is minor. After all, there are other clever people with opposing points of view. So coming to your questions, most people compare the scientific fields and are not into it enough to get the why part. Science has made us expect results which we can see, in a plastic way.
I guess I kind of feel the same. If the reason is that philosophers want to change something they don't like in the world, then I'm pretty sure there are more direct ways to achieve it. So I think "having fun" doing it might be the motivator. Or do they talk about it because it's their job, and a nice one at that? The sad truth is, that I think a major factor is that academics want to get academic recognition. Just like the people in the sciences. It's natural, I don't blame anyone really, but it leads me to believe that what they do is not really important. It gives an interesting cultural background at least.
Coming back: Often times people (including philosophers) argue for a point which they would like to be changed (e.g. a moral or cultural issue) and while trying to find arguments which support their position (which might bring some advantage to their personal situation) they speak as if they it was clear that one could still rely on validity of concepts like necessity in the real world.
I want to emphasise "in the real world" in the last sentence. I don't think that one should confuse the string "box = not karo not" etc. with the concepts of real world necessity, even if that has many applications and captures our intuition.