Hi lsatstudying11!
It seems like you're on the right path in unpacking this passage! I'll break it down first by restating what the two claims are at the outset, and second, describing the author's ensuing criticism.
First, these two claims come in the first paragraph, in lines 5 through 15:
The characterization of this “Chinatown Chinese” as a distinct dialect is based primarily on two claims: first, that it is so different from any other dialect used in China that a person newly arrived from that country might have a difficult time communicating with a Chinese American in San Francisco who speaks nominally the same language as the newcomer, and, second, that no matter which of the traditional Chinese dialects one speaks, one can communicate effectively with other Chinese Americans in San Francisco so long as one is proficient in the uniquely Chinese-American terminologies.
Second, you rightly note that the author takes issue with these two claims. Regarding the first, the author concludes that the "supposed language barrier is, therefore, mostly imaginary" (line 38). One reason the author concludes it is imaginary is because the first claim doesn't account for the fact that often Chinese Americans and newcomers from China can communicate by both speaking traditional Chinese. Regarding the second claim, the author views it at best as an "oversimplification" (line 43) because even that shared vocabulary would only make up a small percentage of words spoken and would "not guarantee mutual intelligibility" (lines 55-56).