Thanks @gpinsky1313. I think that between Tyr, mine and your own collections, we can safely conclude that the drop rates for common and rare weapons are lower (likely half) of the drop rates for the other items of the same rarity.
I suspect that @Turbotarry and @Mkb617 are likely to be suffering from small sample size from their own collections, and so natural variances are masking the decrease in drop rate. I think that only when the collections are large enough (i.e. have sufficient cards opened from chests) that the actual pattern emerges.
At this point it’s a matter of determining whether the model of equal item drop rates or decreased (rare and common) weapon drop rates is more likely to explain the apparent patterns that have manifested it more mature accounts. I think the answer is obvious.
I have already laid out the math above. I invite those who remain contrary to offer an alternative theory on why players might consistently prefer useless non-weapon gear over useless weapon gear.
Note: if you agree with the OP’s conclusions that rare and common weapons drop less frequently than other gear of the same rarity, the main application of this knowledge is to assign a premium to rare and common weapons during challenge rewards or gear purchases, and prefer picking them if given a meaningful choice.