Wikipedia has a lot of information about sign languages spoken around the world and references them as being spoken by deaf people. What I don't understand is, why wouldn't such languages also be used by mute people (people who are unable to produce verbal sounded speech)? (if such were the case, Wikipedia would clearly state such languages would be also spoken by mute people, so what am I missing?)
I guess, you simply misunderstood.
Using spoken language naturally requires both abilities, hearing and talking. Therefore, a person who has difficulty with any of these abilities (or both) is forced to use sign language.
From the medical standpoint, hearing impairment at birth leads to inability to learn how to talk, so such people, even though their vocal tract is fully functional, may have difficulty to reproduce sounds of speech. However, there are modern training techniques simplifying the process of learning.