So I'll be honest. I've never liked the idea of skipping the story we've already got in favor of a new start point. I won't go into detail as to why, but I just think it goes against the idea of the game being an RPG first, and an MMO second.
But there's really no need to compromise that. Ironically a compromise is just what we need.

Here's an idea I just had that I think will future-proof the story for new players somewhat. It's not a perfect solution, but neither is what we have now.

So essentially, I think that you should be able to up-synch your level and gear to an expansion bracket in order to join raid, dungeon, and trial parties that you haven't done the story unlock yet. This would need you to go out of your way to enable it as to avoid potential spoilers for normal story-goers who might want to do some level-appropriate high end content in-between expansions.
There should be a warning when enabling this though. Warnings that you might see things that could reveal story elements before when you're actually supposed to learn them. Ideally giving you the option to limit this to non-mandatory content so you don't end up fighting (Endwalker Trial 2). Still, some players could figure out that one of the lines (P11S) says tie them to another character they may or may not have met yet.

As for the issue of players being given max level toolkits without knowing how to use them, I suggest something similar to Stone, Sky, Sea. You're given the toolkit for your class at whatever expansion level you choose, and a striking dummy. No time limit or health bar. Just the freedom to learn your rotation. There should also be a warning that you should practice a bit before enabling duty up-syncing.
Also, side note. This would help you know if you'll actually like the way a job plays before you invest too much time into it.

These additions would do wonders to get new players into the game much faster. People are already talking about the barrier to entry needed to get into current savage raiding or ultimates for brand new players. The mistake I see being made the most is treating Savage Raiding like it's THE endgame of the game that players are all working towards. It's certainly a big part of endgame, and my favorite aspect of it, but that's the best part of FFXIV. There are many different types of endgame.
You can start doing PvP at level 30, and face the same players who queue in from their level 90 character.
You can start doing Deep Dungeons from level 17.
Glamour prisms can be gotten from level 15, crafting can be started once you got your first class to level 10, and Roleplaying has no minimum level.
I don't see why High End Raiding and Trials should be any different.