Your reasoning is logical as far as it goes, but it has lead you astray. For example a Z boson is its own antiparticle, and so is a Higgs boson. But neither of them travel at the speed of light.
The photon is a boson too, and there's a pattern here. Uncharged bosons have a tendency to be their own antiparticle, while fermions of all types have distinct particles and antiparticles. In theory there could be a type of fermion, called a Majorana fermion, that could be its own antiparticle, but no fermions of this type are known (I believe it's still just possible that neutrinos could be Majorana fermions). Charged particles obviously can't be their own antiparticle because the antiparticle has the opposite charge. So as long as an electron has a charge of -1 it can't possibly be its own antiparticle.
Particles and antiparticles are related by CP symmetry, and a particle is its own antiparticle if it's unchanged by a CP transformation. This has nothing to do with mass or velocity.
In everyday life an electron could never reach the speed of light anyway, because that would require infinite energy. However above the electroweak transition electrons would become massless and would travel at the speed of light. Despite this electrons and positrons would remain as distinct particles.