Sound is a fairly direct way to examine impermanence.
Obviously, a lot more to chanting than that, but if you are looking for reasons to do it, it has all kinds of meditative uses, and it's not something separate from meditation, use of sound IS a form of meditation.
If you want to get more comfortable with sound, there is stuff like sounding "ah" or "hum" as an object of focus, which can be a form of shamatha. Gotta ask..are you sure you just aren't uncomfortable making sounds out loud? When I started with Vajrayana practices I felt the exact same way, I felt really "new age" or something similar and self conscious..felt weird even doing it with other people. Eventually that went away and I now see how incredibly valuable the techniques are, IME sound can transform your thoughts, and have an effect on your mind very quickly.
in addition, things like aspiration prayers, which also used to feel very funny to me, really seem powerful to me now by virtue of the fact that I am saying out loud my wish to achieve something, and saying I will put effort towards it. Personally, this would not have had the same effect I don't think, were it done silently, or just as an abstract thought.
"Just as a lotus does not grow out of a well-levelled soil but from the mire, in the same way the awakening mind
is not born in the hearts of disciples in whom the moisture of attachment has dried up. It grows instead in the hearts of ordinary sentient beings who possess in full the fetters of bondage." -Se Chilbu Choki Gyaltsen