You would be better off knowing one way or the other, and getting a prescription refilled is apparently what your Doctor ordered anyway, and it is a simple way to probe the situation.
If by some chance I have NOT had a renewal properly processed, DESPITE my submission of what they asked and a record of that submission, would they have any responsibility for my coverage due to their lack of notifying me that the coverage was discontinued?
The answer to that is MAYBE but it would be harder to deal with this after you need covered treatment than it is now.
Is there indeed a benefit in leaving the question in limbo in hopes that even if I'm not officially renewed that they would bear some responsibility for not having informed me.
No, there is no benefit to leaving this in limbo.
It they cover the prescription you can probably safely assume that all is well. If they do not, you can get a free consultation from some of the bad faith insurance lawyers listed by location here
I hope this information is helpful.