Agreed - face to face interaction with deities is not handled in 5e or older systems. Maybe it's simply never meant to be, but a card system would be appropriate in lieu of dice.
Well the older systems - 1e & 3.5e - did stat the gods, which I always found kinda ridiculous unless you wanted to run an Illiad type game.
But I wouldn't want there to necessarily be interaction between the gods and mortals. I was thinking more like in the god game you are creating the world or at least the current world state as a negotiation/competition without randomness. Then the PCs on the mortal side - who may or may not be the players in the god game - have to live in that world use the dice based systems of D&D and similar games.
For example if Poseidon really, really wants to punish Odysseus by sending his ship far of course from home he can probably do it. The other gods like Athena would then be able to spend their points helping Odysseus bit by bit on each island they visit since Poseidon had used up his points. PCs playing mortals then play out the adventure on the varied islands Odysseus and crew visited.