I'm really not sure what you're trying to ask with this waffle...?

Exams such as IELTS and CESOL are independent of the schools that have prepared students for said exams in the most part. I work for a Cambridge exam centre, meaning students can sit exams at the school, but we don't mark the papers. After the exam, the papers are officially sealed and go to Cambridge departments for grading. The only certification I award is a grade for their oral fluency, which is again graded during the exam and then sent to the central exam board for processing. Cambridge then grades the exams and issues certification. And I don't examine at my centre, I travel to other cities so there is no chance of me having to grade a student I know or have taught. The same is true for all examiners.

IELTS operates in a similar way.

The vast majority of schools are not accredited exam centres, although a lot are accredited preparation centres.

As the exams and the grades awarded are nothing to do with the school the student has attended, your question is invalid. Students are prepared for a targeted CEFR ability level, the language they learn helps them to pass a fluency exam. If you had any idea of how these exams actually work, or the content of them, you wouldn't be asking such questions. That you are is a clear indicator that you don't know anything about the exams, the bodies that regulate them or what is expected of a student in order to pass them.

And no, quality universities are not worried about the efficacy of the exams as they do understand what is involved in attaining a pass.

Attempting to discuss this topic on this board is akin to discussing particle physics with a reasonably intelligent monkey... You understand the general topic is science but have no idea about what's actually involved. If you really want a discussion about the efficacy of the Cambridge exams, go to Cambridgeenglish.org look at what is involved in an FCE or CAE exam (actually download and look at the sample papers, don't just skim the blurb) and then come back with an informed opinion.