In my personal opinion, I feel that any new players should have to do all of the same questing that each of us have done to get through the MSQs. It seems overwhelming, considering the number of quests, but really it doesn't take that long. I feel that if someone was interested in speeding through it they could finish it within 2 weeks if they focused on MSQ progression. Just my opinion. I halfass derp'd my way through the MSQ and never really felt like it was taking up too much of my time, and I didn't dedicate that much time to it, but I was easily able to clear all MSQs available.
I feel like if you're joining a world, with a story that is to be presented to you, and you are the protagonist, the "Warrior of Light" the defender of the people, at least put forth the effort to play the role. Sure other content is fun, but what are you in a hurry to get to exactly? Running the same 2 dungeons and whatever raids are available for "Allagan Tomes of Whatever" over and over and over and over and over again each week to get rewarded with some gear that you'll be replacing when the next "Allagan Tomes of Whichever" become available making all your hardwork and your gear somewhat obsolete anyway.
At least new players have the ability to bump their ilvls up a lot faster and are more easily able to catch up in that regards. If any MSQs were cut, I would agree that mundane fetch quests could go, but for the vast majority of the MSQs I feel that they should remain and new players should have to make their way through them. It.....really....does....not.....take....that....long. Aside from queue times. I fear queue times for Crystal Tower runs could definitely cause a delay with MSQ progression for newer players, but then again, it seems that there is an unofficial Sunday night CT raid night so there's at least that. Or a new player could grab friends, fc members, ls members, PF peeps, just as many people as possible and make their own groups to run the tower raids.
TL;DR New players should have to quest their way through the MSQs, both ARR and HW, imo. No shortcuts (unless SE removes mundane fetch quests). Sorry, but it doesn't really require that much time and effort and it's probably the most efficient way to level and acquire gear and gil easily when first starting out.