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The Heine-Borel and Bolzano-Weierstrass approaches both used the idea of compactness. But, neither of them used the definition of compactness directly - they both used a theorem that related compactness to some other property. As a refresher:
What does the Heine-Borel theorem say is equivalent to compactness in ? (Hint: the other choice is the Bolzano-Weierstrass theorem)
But instead of using these theorems to characterize compactness, we can actually prove the theorem just using the definition of compactness and the fact that continuous functions map compact sets to compact sets. This is the method we use to prove the more general Extreme Value Theorem for topological spaces.
In fact, this will be the method we use in the unified proof!
There are even other proofs that do not rely on compactness directly at all!