The exact answer to your query depends on the nature of the hiring policy of the concerned organization. Truly speaking, no one can be totally unbiased, and the element of bias is an innate and irrevocable trait of human beings because of the presence of ego in everyone. Mostly, subjective hiring is the result of the subjective mindset of the recruiters.
If BGV is a process post-appointment, its ultimate purpose is the verification of the credentials already furnished by the candidate. It will not be possible for every job seeker to provide all the documents pertaining to their previous employment, and in the absence of one or many, only a circumstantial decision should be taken by the prospective employer based on the genuineness of the available document. If a particular document pertaining to the first appointment could not be verified with the concerned organization which became extinct later, why can't an attempt be made with the organizations subsequently employing the candidate as to whether they verified the fact when the organization was in existence? What prevents a recruiter from conducting the BGV before the issuance of appointment orders? The further continuance or the subsequent closure of an organization is not in the hands of an ex-employee, which is an acceptable fact. If a university falls into the subjective hiring list of the organization, any candidate from such a university can be rejected in the screening stage itself. Just because the candidate has all the potentials for the position to be filled, he was appointed. Therefore, non-existent discrepancies in the BGV cannot be presumed to send such a candidate out.
Better explain these points politely to the organization in a convincing fashion, and still, if they do not comply, you should be bold enough to sue them for damages.