Why do we need tube lemma to prove the compactness of the product of two compact spaces?

Question

I read the proof in Munkres' book Topology which uses the tube lemma but still thinking about an easier proof using basis of product topology : $X \times Y$ has $$\{B_x \times B_y, B_x \times Y, X \times B_y \mid B_x \text{ and } B_y \text{ are basis of } X, Y\}$$ as basis. Since both X and Y are compact, all their finite open sub-covers can be expressed in the same finite fashion using their basis considering that open sets can be derived from all unions of basis. Stating that, it seems easy to me that every open cover of $X \times Y$ has a finite sub-cover which can be derived from basis defined above. Obviously, I'm wrong somewhere but can't figure out where