I think one reason the Pumping Lemma is emphasized so much is that it is a good exercise in logic. We often find that students have difficulties with quantifier

I think one reason the Pumping Lemma is emphasized so much is that it is a good exercise in logic. We often find that students have difficulties with quantifier alternations (there exists x, such that for all y, there exists z, such that...). I think it is a good idea to practise such reasoning (e.g. for complicated module abstractions in software systems). The Pumping Lemma is a good example to exercise such reasoning.