We're deriving intention from wording, not examining it in a vacuum. The wording itself says "weapon." It could have said "any object," it did not. From the deliberate wording choice, we can infer that the ability applies to weapons, and not to anything I have in my pocket. The question then becomes which items in your pocket constitute a weapon; if they all do, there is no point in distinguishing weapons from any other object in the wording.
There is a logical theory that states that if a person lists certain features of a thing, or certain elements of a list, and does not list others, he is excluding them by implication. For example, "Pets permitted in this apartment include cats, dogs, fish, and guinea pigs." The failure to list lizards implies that the list excludes lizards. The argument "But he didn't say we couldn't have lizards" will generally not be seen as effective. In this case, the ability says "weapon." This implies that there are some objects, which are not "weapons," to which the ability does not apply.
We're not talking about what can and cannot be used as an improvised weapon. That's part of the exercise. If we assume that any object can be used as an improvised weapon, and that an improvised weapon qualifies for Quick Draw, we reach the conclusion that Quick Draw applies to every possible object you can store in a container on your person. Since that is clearly incorrect, one of our assumptions - that any object can be used as an improvised weapon, or that an improvised weapon qualifies for Quick Draw - must be false.
We're not looking at what was intended. We're looking at RAW. The RAW is that you can draw a weapon. The assumptions are that you can draw an improvised weapon, and that any object can constitute an improvised weapon. We are determining whether these assumptions are accurate. Again, you're arguing for lizards, as above. Stop that.
Correction. We are following logic. If we assume that any object can be used as an improvised weapon, and that an improvised weapon qualifies for Quick Draw, then what we draw is irrelevant - it all qualifies. I use the term "any object" because if an improvised weapon can be composed of anything, then anything should be able to qualify for Quick Draw, if and only if our two assumptions are true.
The question, therefore, is which assumption, if not both, is false?
Can any object qualify as an improvised weapon? Does anybody have RAW on this point? If any object can qualify as an improvised weapon, then the only falsifiable assumption is that improvised weapons qualify for Quick Draw.
I think the argument hinges, however, upon the second assumption: Whether an improvised weapon can qualify for Quick Draw. If it can, we are limited only by the definition of "improvised weapon." If it cannot, however, the entire logical sequence collapses, which frankly makes more sense to me.