One part of the Time player's role is to be able to initiate a Scratch in the case of failure, and another part may simply be that it is the complement to Space, and duality is a strong theme throughout SBURB in general. Otherwise, I'm not sure.

Yes but the Scratch is not a necessary element, in fact it’s an extremely dramatic emergency reset button. Complementing Space is a thing that could happen, but the Space Player is needed for the Frogs specifically, so I’d assume the Time Player would be required for something physical as well. So far the thing that seems the most likely is that they’re a safety net to fix up mistakes and give more time to the Frog Breeding and other important things- But even then that raises the question, if you were to have a FLAWLESS Session, would you be able to win without a Time Player?

It’s also possible Echidna does not want to light the forge unless it’s for Hephaestus, in which case that would indeed require a Time Player, and furthermore, one whose Denizen hasn’t been replaced with Yolobroth or Abracadabra-man.