I think OP has a valid question, even if I agree with everyone saying no. And it's understandable if they're someone who either cares only for gameplay or is an alt-oholic. [EDIT: forgot to check how many pages a thread has a again and ended up reacting to first page comments, oops]

That said though, the whole point of this game is to get into the lore and actually learn about the world the devs put so much effort into building. This isn't WoW where you can just bumble your way around while knowing jack squat (and even WoW only unlocks pugging for specific dungeons after you progress a zone's story to that point; of course you can still just waltz in through any dungeon entrance if you find them, iirc). FFXIV isn't WoW though and I appreciate it making everyone do the story so there is little to no confusion on the lore at all. *shudders at every Alliance vs. Horde debate that happens because of few people being both minimally biased AND having the full picture*

So, no, it wouldn't be feasible at all because the FFXIV team WANTS you to experience the story properly. The level skip potions are mainly for people with alts who don't want to have to repeat everything they've already done, but if someone wants to skip story badly enough to fork actual cash they're not stopping them either. But you are paying extra if you want to "cheat" and not play the game as it's intended to be played.