This is a pretty common misconception of path integrals from how Feynman explains them to a non-physicist audience. The path integral formulation cannot work from a particle-based approach, where you integrate over all possible trajectories of electrons, photons... . Instead, you are actually integrating over all possible configurations of fields, the electron field, the photon field... . As far as I know, there is no way to view the universe as being fundamentally made of particles that is compatible with our modern understanding of QFTs.

Also, the path integral really shouldn't be taken too literally. It's not reality, it's just an equation.