Well, you don't have to do the story at all anymore. Once adventure is unlocked, you can start a character in adventure mode and hit act v at level 1 if you want. Everything scales to your level. The entire map is open to you. Every act has bounty objectives and at any moment, one act is designated as having a bonus, so if you are doing bounties (a mix of kill this boss, complete this event, and kill all enemies at X location) you will typically go where the bonus is. The bounties reward you with recipes, crafting mats, and the occasional legendary.
But to get back to your question, nightmare and hell difficulties have been replaced with a much less coarse system. It now goes normal, hard, expert, master, and then torment. The latter has 13 separate difficulty levels. I suppose if you like replaying story mode, you'd start a "nightmare" replay at Torment 1 and just keep bumping the difficulty as needed. I can't imagine doing that, though. About the only thing I will miss from the story is the character of the moment saying "Spider Queen?" Before the fight with Arenaea.
Moreover, these days, it's apparently possible to level to 70 in 2-3 hours if you are good at speed farming and willing to grind (I'm not). The season journey gives you a bunch of stuff to do, leading you along the character progression as you go. It probably took me more like 10 hours to get to 70 but I enjoyed it.
I think the thing I would suggest is to go beat act V to unlock adventure mode if you haven't already. Then start a season character in adventure mode at level 1. Pick a class whose season set matches your playstyle:
Here are the sets granted by Haedrig’s Gift in Season 9:
Barbarian – Might of the Earth
Crusader – Thorns of the Invoker
Demon Hunter – The Shadow’s Mantle
Monk – Monkey King’s Garb
Witch Doctor – Raiment of the Jade Harvester
Wizard – Firebird’s Finery
The first four chapters of the season journey will lead you through level 70 and all the way to Torment 4 difficulty. By the end you will have one of the 6 piece sets listed above (open all of the gifts from completing chapters 2-4 on the same character!). That point marks a nice finishing point, or you can continue on to the more advanced parts of the journey if you like.