I wonder sometimes what makes the player think they know more on the game than the people who created it. It's as if an on-looker tells the artist what he was trying to paint. It's ignorant. The artist painted something entirely different than what the person said. It's the same thing with a reader and an author. The author of a book may put in or write a book that some see as a satire or some other thing, when the author simply wanted to write what they wanted. It doesn't have to make sense to you yet, especially when the TES creators and writers haven't completed the TES story and lore.
It is true you may find some holes, but sometimes you just make it bigger than what it really is. We can't expect an explanation for every stone left unturned or every blade of grass left uncut.
It is true that only a Septim can wear the amulet. You prove that in the game by your inability to wear it. Perhaps the Septim line is a branch of the Reman line. After all, weren't there cousins and etc who ruled? What about a female (I do not know, asking the lore-ists)? If there was a female, then she would most likely take the male's last name. And in that, it could have been turned into Septim.
It is true M.C. wears the amulet without a proper explanation. Perhaps there's more to it than we understand. Maybe we won't know till the end of the TES legacy. After all, there's years till it comes to an end. A writer doesn't give you everything at once, they keep you guessing.
I first played Morrowind when it came out. Even after finishing the game I didn't understand a lot (Since I didn't know the lore before). And then Oblivion came. And, in this vast legacy, I found most, if not all, the pieces to fit without an understanding of the complete lore. That doesn't mean I didn't second-guess things, I did. But in that, it made sense to me, because my knowledge was virgin and untainted by the past lore, which would have just clouded my judgment. Even to this day, with an incredibly large understanding and knowledge of the lore, I still feel the same way. I don't see many holes. Of course, I'm expecting some posts or thoughts of "Idiot..." because I say this. But plank before sawdust, my friend. Perhaps it is I who sees more. Or perhaps I see what I want to. We will never know till this comes to an end.
But finally, I ask that you, instead of making holes or finding them, try to piece it together instead. Try to make it fit, instead of tearing it down. Maybe you'll enlighten yourself to something new. Go in with an open-mind, not a closed-one.
I agree with the spirit of your post.
The amulet seems to be a legitimate issue, though. It would have been fine if only in-world sources (i.e., characters and books) said it, because they could just be wrong. What makes it problematic is that you (the player) couldn't wear it, and when you tried you got a prompt that says only a Septim can wear it (or something like that, I forget exactly what it says), and there's a loading screen that suggests this as well. But neither of these (prompt or loading screen) seems properly in-world, and the fact that the player can't wear it would suggest that there is at least some requirement.
Unless someone wants to argue that loading screens and prompts are in-world phenomena (which I doubt), the only way to overcome Mankar wearing the amulet without retconning is to say Mankar has Septim blood.
That said, I concur with the spirit of trying to fit things together rather than presuming the holes are real and without answer.