A skip to get to 70, probably. To 80, almost certainly not. Skip potions are offered to allow players the option of skipping over older content to reach the latest expansion. They're not for skipping the latest current content.
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Adding to the "things the story skip does/doesn't do", I found a quirk in that: the summoner job quests require you to have beaten the 'easy mode' primals before you can learn to summon their egis. Skipping means that from a story perspective, your character has already done that, but the game doesn't recognise it until you've actually cleared the battle.
I assume you'd be able to buy a "skip to Lv70" book, but it will cost you the same whether your character is Lv60 or Lv1.
I mean, from a story perspective, the skip has moved you to a point in the story where your character has already fought those primals and should be ready to learn to summon them. But in the game programming, there is no record of you-the-player having completed the battle instance, so it won't let you do the summoner quest until you've actually completed the trial.
I realised tonight though - I actually think the story skip has reset the record of my progress in all dungeons. I got Tam-Tara for leveling roulette and the map was blank, as if I'd never been in there before, even though I was past that point when I used the skip.