Yeah, that's how it is usually introduced to students. First you introduce them to finite state automatons, then you show them a cool trick of extending the words by walking in circles on the automaton's state graph, and only then you mention that this is basically a pumping lemma. After everyone understood the point of the pumping lemma, you write it down formally using five quantifiers, so that student can write it down concisely, as the idea is already understood at that point.

I agree that the formal statement of the pumping lemma can be very uninspiring, but it only hints to two important facts. First, it's very important to have a good teacher, who is able to introduce ideas in a way and order they work for you. Second is that in math, it's the proofs and ideas that are important, not theorem statements.