I do not think this is correct, at least for most of the sub-fields of computer science and physics. Prestigious conferences and journals invite only well known researchers to review, who are mostly professors. Few post-docs get invited. Each paper gets multiple reviews (3-5), and conflicting reviews get discussed making the review process quite rigorous.

Reviewing a paper is mostly charity and gets you nothing in return, except the fact that when you submit a paper, someone will do the same. It does not help in the Green Card process at all, or boosts your academic credentials.

Regarding overworked professors, they are allowed to delegate reviews internally to students to ease the time they have to spend.