What other people have said about having the choice to remove the processor or leave it on- a CI recipient will always be considered deaf, from an audiological perspective.
I did my high school Senior project on Deaf culture, and read "Deaf Like Me". It's very understandable why the Deaf community felt the way they did with regard to the matter of trying to teach kids how to talk, and their initial reaction with doing that with CI's. Most of the methods that were used on children with profound deafness, such as trying to feel vibrations that the throat made as a way to trying to learn to "hear" sounds towards speaking them, simply did not work and were a waste of time. That kind of thing is what some members of the Deaf community remember. They may feel triggered and for that I can't blame them.
However, with CI's learning to hear/talk is not such torture. The technology and sound quality that we have today with CI's is a far cry from what was forced on children with profound hearing loss decades ago, as I can confirm as well as many other CI users. Also, many more early childhood education centers and preschools are using play-based curriculums & environments to teach skills in all domains of development, which is most appropriate for young children in their learning.
Babies' brains are open to all of the nuances of ALL languages until 6 months. That's amazing. However, after 6 months, their brains will become most attuned to those languages that are being used with them in the home. So if both English & ASL is being used, I would expect that those kids would learn and process the two very naturally. There is no reason why children with CI's can't become bilingual in ASL and spoken English. They are both languages and as long as the child is getting exposure to spoken English, he/she will develop it. As long as they are getting exposure to spoken language, exposure to ASL will not hurt what oral language the child is already exposed to. What matters is consistency. After all, we sign with hearing babies, so why would signing with deaf children who receive CI's hurt their progress?
You can look at kids who immigrated here at a young age, with their families, who were fluent in their native language, initially. I grew up and went to school with several of those kinds of kids. The kids learned English pretty quickly, within a year or so. It never hurt their fluency in their native language. As to the parents, some were successful and some weren't in learning English but it was much harder for those parents than their kids. And that has to do more with brain plasticity than anything else. So if kids are not given CI's at a young age, that detracts from the opportunity to have the same kind of progress that they otherwise could have had.