Character creators generally run down two paths. Either you are stuck watching somebody elses story or you have a generic story regardless of your choices.
In the 16 bit era we didn't have anything like character creators and we were doing great. The characters made the storyline. Look at Cecil from FF4. Here we have an awesome dark knight who has it all and loses it, then becomes a paladin and that's where his story really begins. FF6 had no clear main character that you start and finish the game with. Terra is the first one you play with, but you don't even have to pick her up in the world of ruin for the final confrontation. About the most customization you could get in this era was a few games you could choose between a male and female protagonist. Sometimes this had large impacts on the storyline, sometimes not so large.
Playstation era had a few games like this. Star Ocean 2 is a great example of a drastic change in story based on who you start the game as. On the PC front we had some of the first games where we could customize our character such as baldurs gate and Icewind dale. Because of this, we found that we were playing the story with the other 5 characters in the party while your main character was just the one who made it game over when he died. Icewind dale had some interesting dialog options if you chose certain classes (like bard), but it was rather lacking in characters because you created your entire party. In many ways, this was a great plus. It was also one of the last games where you could roll your entire party rather than some lame weighted points system (Since any stat over 14 costs more than one point, why would you do anything other than make your character have 14's down the board?).
In the PS2/Xbox gen we really started to have some decent character creators such as Morrowind, KotOR 1/2, etc. I love these games. But I also enjoy games like Shadow Hearts, Front Mission 4, Disgaea and other great rpgs of that era. This gen also introduced mainly voice acted dialog. Once you are able to customize so very much the only thing left is the voice. Mass Effect did a pretty good job of being able to fully customize what your character looked like, but you pretty much sounded like the same guy or bitchy woman however you looked. The other option is to make your protagonist silent. The only game I know that has an excuse to have a silent protagonist is Drakenguard.
My answer is they don't absolutely neeed a character creator. If you create a solid and enjoyable character for me to play, I'm totally into playing said character.