Why does lithium fuse at lower temperatures than hydrogen?
The slowest reaction rate in the pp chain determines how quickly hydrogen can "burn" in the core of a sun-like star. That rate-determining step is actually the fusion of two protons to form deuterium via the diproton and a weak interaction decay.

The fusion of lithium, whereby it fuses with a proton and then splits into two Helium nuclei is actually part of the PPII series of reactions that convert helium 3 into helium 4. It does not involve weak interactions and therefore the cross-section at a given temperature is much higher than the first step in the pp chain, regardless of the stronger coulomb repulsion between the reactants. It is therefore initiated at lower temperatures - as you say, Li "burning" occurs in low-mass pre main sequence stars well before hydrogen "burning" commences (but after deuterium burning).