What I liked about both Morrowind and Oblivion is the fact that you are NOT a hero.
At the beginning you have no idea who the hell you are at all, and in Morrowind you get the feeling that other people have also no idea - they ask your name and so on.
In Oblivion you even never get to know why you are in jail. You have no past and its great, because you can just imagine it. But you can become a hero - if you want.
I found it great that people in Morrowind seemed to think that the world can take care of itself and let you do what you want. Somehow it gives you a feeling of absolute reality and I enjoyed this freedom a lot.
Its a pity that it was not always so in Oblivion. Ever again: you MUST do it NOW! And escorting people was a real pain in ... all parts, armor included

I don't like being THE hero. I wouldn't mind if being THE hero was an option (among others). I'd personally like there to be multiple avenues to a final solution (or like others have suggested, at least multiple reasons for me choosing a single avenue). Multiple avenues happening at the same time, forcing the PC to choose one while significant/essential NPCs handled the others would be pretty sweet in my books. Then the necessary praise would be hung on someone else's shoulders and I could go about my game with NPCs making reference to some tangible NPC having done something marvelous while I contributed in another way. In my mind, that would be much better than me having to become the Hero of Kvatch and Champion of Cyrodiil just to close the gates and see Tamriel at peace.

I understand the benefit of forcing the PC to do everything so you can simply say the CoC (or Nerevarine) was essential, blah blah blah, and then write them off, but I think you could do that with a NPC as well.

Something along the lines of:

I deliver the amulet to Jauffre and spill my story.
He says where the heir is and asks me if I'm able to find him.
At this point I have a multitude of options. (I can say no, telling him to do it himself. I can politely decline, citing my injuries or inexperience. I can ask what's in it for me. I can offer to be a part of a team. I can boldly say I will go by myself)
If I decline, he sends someone else to do it (Someone tangible - NPCs, not some scripted event. I then have the option to have a change of heart and tag along, but the honor will fall on the other person's shoulders. Or I can let them go on their merry way).
If I say yes, then the quest is left to me to finish at my own pace. No time constraints. But if I say no, then I don't have to worry that the game won't progress without my efforts.

Granted, if you say no then good luck being a part of the team that leads Marting back to CRT, but that's a consequence. But being inserted into the quest line would simply be through rumors or revisiting WP and asking around, etc...

It's not a complete solution, just something I wouldn't mind toying with.