I think that it might be better if everyone had to play through the Story mode for their first time playing a new seasonal character at the start of a new season. It would make it feel a lot more like a fresh "reset", instead of just spam running bounties or whatever the "fastest" way to level is.
That said, I think it would be more important to give more incentive to make a seasonal character in future seasons. Because the rewards for it seem sort of underwhelming to me.