Flat-band systems such as magic-angle twisted bilayer graphene host strong-correlation superconductivity at vanishingly weak coupling, yet how quantum geometry and pairing fluctuations conspire to drive this phenomenon remains an open question. We investigate finite-temperature superfluidity in a quasi-two-dimensional Lieb lattice using a pairing fluctuation theory with a band-uniform attractive interaction g<0 that isolates the intrinsic quantum geometric contributions. Quantum geometry significantly amplifies superfluidity; the geometric pair hopping integral surpasses its conventional counterpart, and the geometric superfluid density becomes the dominant in-plane transport component. When the Fermi level enters the flat band, the BCS paradigm breaks down entirely; the pairing gap and Tc shift from exponential to anomalous power-law scaling Δ, Tc ∝ |g|^ν (ν>1), and the superfluid density inherits an unconventional power-law temperature dependence at low temperatures. In the 2D limit (t_z=0), the pseudogap at Tc nearly saturates the zero-temperature gap even at |g|/t=0.001, placing the system in a strong-pseudogap regime that would otherwise require unitary or BEC-scale interactions. These findings establish a microscopic mechanism for flat-band enhanced superfluidity and offer testable predictions for ultracold atom experiments.