While I do think this is an interesting idea, I'm going to play devil's advocate and point out the biggest problem it doesn't solve which is a big convenience the potion brings:
Unless Square MASSIVELY boosts exp gained from 1-50 and 50-60 stuff, people will still have to go through a greater portion of the MSQ just to reach the per-requisite levels for your idea. Or they can zerg the Palace of the dead, which they may not even be interested in, or could very well be dead in Stormblood as all the current players move onto the expansion content.
and subjecting new players to massive fate grinds isn't a great way to entrance players into staying either.
Because in reality, dungeons are heavily gated by the MSQ, that by the time they reach 50/60, the most benefit they'll receive is skipping the patch content. Then what if their favorite job they want to play happens to be DRK/AST/MCH? Bam, they now have to spend even more time getting it back to 50 and up to 60 to reach Stormblood's entry level in your idea. I think for your idea to work as well as conveniently a a jump potion would solve, 1-60 would need to take very little time to achieve. Because lets face it. People who buy the jump potion are interested in one major thing: The newest expansion where the massive majority of players are currently doing. and the longer it takes for them to reach it, the less likely they will be to stick around and provide their monthly sub to Square.
For as much as people like to hate on it, WoW is king when it comes to this accessibility. I could go out and purchase a copy of Legion and be at the same expansion as everyone else, whether I choose to level normally or use the skip, doing the same content the massive majority of the playerbase is also doing with very minimal effort. If Square wants new player retention to be high, it'll either have to make compromises in how it designs its MSQ, or they create a jump potion.