No, you don't have to clear coil for any mechanical reason, but coil in particular has a very good story, and a new player might want to experience that. Extreme primals are generally a tougher argument to make, since the only practical benefit are the mounts and possibly weapons for glamour purposes, but some people might just want to complete them for the sake of doing so. Square Enix doesn't want to have content that is literally impossible for a new player to ever clear. No one wants this to become FFXI where no one had any incentive to do content with you so if you were late to the party you were basically screwed. (Actually scratch that, I think a lot of people here want exactly that, but I don't!)
I actually don't think the difficulty gap is as extreme as you present it, at least not for primals. Certainly there are terrible players that can barely clear a basic dungeon, and should be nowhere near extremes. But most of the "skill" involved in completing the hardest content is just learning the fight. Of course someone who's never done the fight is going to be worse at it, but given teaching, you can get most decent players through an extreme. (I've never actually done savage or original coil pre-echo, so I'm not going to presume to comment on how hard they actually are.) That's not a skill gap, it's an experience gap; indeed, the fact that it's an experience gap is why an "I've already cleared this content" button is useful. There are plenty of people without clears who are much better players than some people who've been carried to a clear.
More important than any of that, though, is that my point is not that new players should be able to "slip into old content queues and get carried." It's that if you give players the option to exclude new players, they will take that option, thus reducing the pool of players available to help them get those important first clears. You need to look at this problem deeper than just on the surface.