Like left-handers, another group that isn't much of a group, stutterers occur fairly randomly across the population. So, the vast majority of their social allegiances are to non-stutterers. And most stutterers have non-stuttering loved ones.
Queen Elizabeth II believes her father's stutter was somehow related to his being forced as a child to switch from left-handedness to right-handedness, although the scientific evidence on this theory remains vague, at best. But that raises the point that Left Handers Liberation -- the big social change in the first half of the 20th Century when parents stopped forcing their lefthanded children to write righthanded (Ronald Reagan was a natural lefthander switched over to righthandedness, while three of the four subsequent Presidents have been public lefthanders) -- is completely off the radar. PBS never runs documentaries celebrating this triumph over bigotry.
Lefthanders tend to view themselves as slightly better than righthanders (certainly at baseball, perhaps at creativity) who have to put up with a lot of hassles dealing with the physical world. In general, however, society expects lefthanders to pay for their own accommodations. For example, a running joke on The Simpsons is that Ned Flanders owns a shop at the mall that sells expensive scissors and so forth for lefthanders. Left-handed golf clubs are rare, so most left-handers play golf right-handed. Nobody ever gets worked up over this bit of unfairness.
Stutterers don't particularly want "awareness," either. Most stutterers would like to stop being stutterers. They view stuttering as a defect, which lefthanders generally don't view left-handedness.
Stutterers aren't, on the whole, all that articulate in speech (although there's a subset of stammering as an affectation: e.g., Anthony Blanche in Brideshead Revisited or William F. Buckley going "B-b-b-b-b-b-but" on Firing Line as Hubert Humphrey steams onward).
It's worth comparing two kinds of deaf people: those who start out deaf and thus learn sign language, and those who gradually go deaf. The first form a small but rather fierce identity politics group, since their primary language is signing. (American signers can converse easily with French signers, but not with English signers, because American Sign Language is an offshoot of French Sign Language.) So, they form an insular cultural/language community. Innovations like cochlear implants threaten to take people out of the community, so leaders of the sign language using community tend to be against them. There is a lot of radical deaf activism at the sign language college, Gallaudet University.
On the other hand, the profoundly deaf aren't at all articulate, neither in speaking (of course), nor particularly in writing -- they have to think in sign language, in pictures of fingers according to Oliver Sacks, then translate into English. So, the media mostly ignore them.
People who gradually go deaf, on the other hand, do not form much of any kind of identity politics group at all. They are very happy to get better hearing aids.