Despite them attempting to do it, doctors have not yet found a cure fo

It's actually a grammar thing! Because attempting is being used as a gerund here (an -ing verbal used as a noun, not as a modifier or as part of a verb tense), them would actually need to be the possessive their for (C) to work. This is something that the GMAT doesn't feature in questions too often; that is, there aren't a lot of cases in which the right answer has a possessive + gerund structure. Instead, the GMAT will usually find a way to go with a clause structure and actual subject + verb, but there are definitely more than a few questions in which you can use this possessive + gerund principle to eliminate 1-2 answer choices. Here are a few more examples of incorrect and corrected versions of this principle:

I don't mind him coming to the party. Incorrect
I don't mind his coming to the party. Correct
I don't mind that he is coming to the party. Correct (with GMAT-preferred clause structure)

In spite of Sarah being a good person, she keeps losing jobs. Incorrect
In spite of Sarah's being a good person, she keeps losing jobs. Correct
Although Sarah is a good person, she keeps losing jobs. Correct (with GMAT-preferred clause structure)

It's tricky because (C) represents how a lot of people would actually speak this phrase, so our ear will tell us it's not too bad. The GMAT loves to take advantage of things like that Also fun fact: rumor has it that part of the reason that Van Gogh removed his own ear was that it kept getting him wrong answers on the GMAT. Tell your friends!