連…都… raises one extreme case on its own. 別說…連…也… stacks two: 別說 concedes the lesser-surprise case, 連…也 tops it. Drop 別說 and you lose the comparison, keeping only the climax.
Prohibitive 別 stops an action: 別說話 (don't talk). In 別說 here, 別說 means 'let alone / never mind' — it raises a case to dismiss as obvious, not a thing forbidden.
✗ 別說老師不認識，學生也不認識。 → ✓ 別說學生不認識，老師也不認識。 (the harder, more surprising case goes after 連…也, not after 別說)
✗ 別說學生不認識這個字，連老師都認識。 → ✓ 別說學生不認識這個字，老師也不認識。 (one predicate runs across both rungs; the 連…也 case keeps the same negation)
✗ 別說他來，也我來。 → ✓ 別說他來，連我也來。 (the second rung needs 連 to mark it before 也)
English 'let alone' runs the strong case first, weak case second ("can't run, let alone walk"). Mandarin runs it the other way: 別說 takes the milder case, 連…也 takes the harder. The rungs are flipped from the English order.