Long post incoming, but here's my suggestion.

I think they should:

1 - Reduce the amount of "be a delivery boy/girl" thing on mainstory. There's a lot of moments on MSQ that you have to keep going in and out of a zone countless times just to chat (Hello Quicksand) or even worse when you have NPCs standing next to each other but for some reason they can't both talk to you during the same action. Just cut it out an put it on a skip-able cutscene or make it like that recent sidequest where you unlock the Haurchefant emote (the summary part of it, not the delivery boy one). Also, give more experience on MSQ up to 50.

2 - Add a "Narrator" system where you can set MSQ to be converted into a "long" cutscene in case you have more levels on your character than what's needed for the quest, or in case you're too ahead of gaming experience compared to where your MSQ is. How this would work? Let's say you're leveling with all your neat friendship items and boots and etc, and you're at level 26 but your MSQ is still asking you to go report your progress at the Copperbell Mines. This system would ask you if you want to watch the cutscene that summs up what happens in between that duty and the next one, to the point you unlock The Bowl of Embers and can progress on it, instantly rewarding you the exp you would have gained if you did the mainstory only instead of whatever you did before (like gettin 2-3 levels on a single dungeon, or doing sidequests). It would still gate players behind a "you have to play the game" but it would make it much less "mainquest grindy" in a way that you wouldn't feel like going back all the way to the whole "deliver this" if you could be already doing something else MSQ-related.

3 - After 50, add that system to the way you get ilvl up to Before the Fall. You would then get a text like It seems like you have enough item level for the next step of Main Scenario duty, would you like to jump to your next duty and watch a summary of the events in between? This way if you're already familiar to the game or have friends that can help you get your ilvl 120-ish gear, or if you have enough gil for it, you would go straight from, let's say The Praetorium to Moogle Hard Mode while watching a cutscene that tells you what happened for you to get there.

Of course this would be a feature that only affects A Realm Reborn, because it's meant to be a catch up system to Heavensward and upcoming expansions, meaning that upon reaching Before the Fall patch you would stop being able to jump mainstory quests, being required to play it the "normal" way from 50 to 60.

This system would still lock players on those small storyquest duties, sort of "forcing" a basic knowledge of how your class works and also allows players that want to skip the mainstory to be able to by just accepting the cutscene summary and then skipping it like any other cutscene in the game. If you want to enjoy the story, you can. If you want to just play the game (while still being gated behind the small list of MSQ duties), you also can.

This is just an idea that I think might help, and that would also benefit people that aren't comfortable with HEALER/TANK roles on a first run a lot.
- On the healer/tank perspective: You already have a really fast que, so experience and time avaliable for it isn't a problem. You can simply run a dungeon twice while trying to get through content and easily get 4 to 5 levels, making the skipping eficient.
- On the DPS perspective: You have long ques, so you'll probably like to do your sidequests for more exp and levels while you wait on your duty, but since it takes so long you might get a lot of extra levels in the process, and even though it's nice, it can be very grindy to do a MSQ with a lot of small steps when you're 5 levels ahead of it.

I think it's a win/win situation, and could appeal to both new and old players, since there's a lot of different ways to level on your first run to 50 and most of the time the Mainstory quest gets far behind of your character's level.