Understanding the pumping lemma is essential to really understanding why regular languages are limited. Which in the real world is important for quickly assessing the question of "can I hack this solution together with some clever regexps or do I need a real parser?"
I'll agree that the y-combinator is less essential, however if you even have a sense of what's going on it means that you have an understanding of the basic framework of functional programming and a minimal grasp of the lambda calculus
Not really. You can understand why regular languages are limited with a simple counting argument; the pumping lemma follows on from that relatively easily, but it doesn't really add much understanding IMO.
The pumping lemma is cool, but the Myhill-Nerode theorem is more powerful (as someone mentioned).
I think, too, that what was meant by 'contrived' was that the pumping lemma is not really an equation. But you don't get hits if you talk about the top ten theorems CS geeks should know...
I'll agree that the y-combinator is less essential, however if you even have a sense of what's going on it means that you have an understanding of the basic framework of functional programming and a minimal grasp of the lambda calculus