How does hyperconjugation lead to stability?
What you are ignoring is the magnitude of the effect.

While hyperconjugation does decrease the electron density of a localized bond and slightly increases the energy, the overlap with a high-energy, empty orbital is at least very poor. Think that the orbitals are very far from each other and have different symmetries. The total electron density of this bond would remain largely unaffected.

On the other hand, the difference between zero electrons and partial electron density for an empty orbital of high energy is significant, and it increases (to a moderate extent) the stability of, say, a carbocation. Besides, a single orbital often receives electron density by hypercongujation with more than one bond.