Official Explanation Magoosh :
After the underlined phrase we have "… and who yet inspired …," another "who" clause in parallel. We might expect at least one "who" clause to be in parallel with this. Technically, another kind of noun modifier might be in parallel with a "who" clause modifier. It will depend on the exact construction.
In the choice (A) version of the sentence, we have "who was … and who unwittingly destroyed … and who yet inspired …" This is the somewhat awkward parallel structure of "X and Y and Z," rather than the more natural "X, Y, and Z." Also, this has the very unusual tense of past perfect progressive "had been hoping"---there is no reason for this to be progressive. Choice (A) is incorrect.
Choice (B) provides no first "who" clause to be in parallel with the one after the underlined section. The absolute phrases noun modifiers parallel to the "who" clause noun modifier is technically correct but less than ideal. Also, it makes perfect sense to say that Schliemann "unwittingly destroyed" the artifacts, but it is awkward to say that he was "the unwitting destroyer" of the artifact. The "unwitting" aspect and the "destroying" pertain to one action, not to who the person was. Choice (B) is wrong.
Choice (C) is clear, with no grammar or logic errors. This is a promising choice.
Choice (D) begins with an redundancy: "cite as an example." The GMAT is never fond of redundancy. This create false parallelism, mechanically putting every single verb into parallel with no regard for the logical relationships. Choice (D) is wrong.
In Choice (E), the "when" + [participle] structure is questionable. The big problem is the semicolon break. A colon would work better, but a semicolon creates too much of a divide between the ideas in the first and second halves.
To explain further: the problem with the "when + participle" construction in this answer is not the construction itself, but the context that it occurs in. In Choice (E), "when deploying" occurs within an independent clause (after the semi-colon). This disconnects "when deploying" from the parallelism (the who that is doing the deploying).
Choice (C), however, does not have the "when + participle" construction in a separate clause. Thus, it forms an appropriate parallelism that makes clear, logical sense. In other words, "when deploying" is questionable after the semi-colon, as in Choice (E), but is perfectly clear and acceptable in Choice (C).
Thus, choice (C) is the best answer.