A character’s nose shouldn’t affect the plot. Just saying.
As you can tell, I’m a plot person. I am more invested with character personality, character development and the emotions or memories that they managed to evoke from within me. I feel that CC takes away from the story. I like to read a story and see how the author envisioned the characters. I can be immersed in the story and feel that I am living in the story regardless of CC options.
Another reason is, not all the MCs acts the same way or feel the same way as the readers do so when the MCs do certain choices that is opposite of what I usually would have done is a bit off-putting to read. For example, if the MC is pro-abortion and I am pro-life, and there is no options to choose from and I had customized my MC, it is a bit hard to swallow and read. I can understand that the readers are playing as the MC and not as themselves. Sometimes I find the MC in scenarios that are extremely uncomfortable or I’m reading scenes that deal with sensitive topics. To read those scenes with CC is well… you get the point. That is one of the reason why I tend to skip CC options.
I think the best would be putting CC but including a skip CC option too. Put an option, “This looks great/This is perfect” etc for those that wish to skip the CC entirely and just get on with the story.
You cater to both groups that wants CC and to those that don’t care about CC.