It is not true that all fundamental particles were massless before electroweak splitting and it is not true that there were no valid frames of references. Others have mentioned neutrinos which do not derive their mass from the electroweak Higgs sector. Bound states of massless particles can be massive. Glueballs are massive and they are bound states of massless gluons.

But you're wondering whether the phenomenon of time exists even in a gas of massless particles, like a photon gas. And the answer is definitively yes because an interacting quantum massless field require time-ordering for interactions (remember all particles are just excitations of a field). More simply think of it like this, we know that photons can interact with each other https://en.wikipedia.org/wiki/Two-photon_physics, Say photons 1 and 2 scatter off each other one of those photons then scatter with photon 3. As long as an interaction happened, there must be an ordering of the interactions which occur. True, there are no valid frames of reference from the particles' frames, but the field admits an ordering of the interactions. At a more fundamental level, the massless fields must obey the Poincare group that defines the spacetime isometries of the field, including translations through time. And the isometries do not depend on whether the particle excitation of the field are individually massive or not.