If I understand your question, I think both your assumptions are wrong. In the first case, I believe we have multiple teenagers but not multiple diet attempts. I further believe "at that time" is not so much meant to exclude other times (possibly implying they were overweight at other times) as to point out the limitations of the scientific data.
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I read that when present perfect is used, then specific time can't be mentioned.In the same clause. That doesn't apply here. There is no specific time associated with the attempts, which is in a present perfect subordinate clause -- but with the idea of being overweight, which is in the main clause, which is not in the present perfect. Still, to my ear, it sounds