Free speech is crucial for people to express themselves but sometimes it must have its own limit so that the people wont simply spread false message etc . Free speech still can be practice but it will be limited so that the negative impacts from the people's word can be lessen
If you go into a restaurant that prohibits use of profanity, that restaurant owner isn't affecting your inherent right to free speech. You can always go to another restaurant if you want to use profanity. If I ask you not to carry your gun in my home (which I never would) you can choose not to come over, but you still have the right to have and bear arms.
Yes, free speech can co-exist with internet filters, because there can be legitimate limitations on free speech. We have decided as a society that we don't want children exposed to certain things. That is fair, but it also does not mean that people cannot say what they want to. We can have both.
Just because we have free speech in the United States does not mean that we are allowed to say anything to anyone at any time. So just as we can not slander someone or scream fire in a public place when there is none, so we can not just put anything we want on the internet.
Free speech exists without Internet filters all the time. How did people have free speech before the Internet? It's the same way we have free speech now since we can say almost anything in the United States. Anyone can say something in an Internet forum regardless of the filters in place.
Free speech can exist with Internet filters, but it will be limited. With every internet filter that may come, free speech will be reduced, but it is still there. It is possible to co-exist if a huge filter is not placed on issues that deal with democracy and things related to freedom of speech.
There are some morally viable reasons for restricting the Internet. However, those actions deemed immoral are not only restricted in access, but publishing. Publishing is where the focus should be. Web filters are typically an abuse of power. The restrictive access is automated and will not be effective anyhow. There are ways to subvert the filters. Free speech will find its way through internet filters.