We give a new, conceptually simpler proof of the fact that knots in S3 with positive L-space surgeries are fibered and strongly quasipositive. Our motivation for doing so is that this new proof uses comparatively little Heegaard Floer-specific machinery and can thus be translated to other forms of Floer homology. We carried this out for instanton Floer homology in our recent article "Instantons and L-space surgeries", and used it to generalize Kronheimer and Mrowka's results on SU(2) representations of fundamental groups of Dehn surgeries.