The core idea of this root was:
It’s the origin of shove and shuffle (which was a “quick, repeated shoving”) and also shy, which is based on a sense of “being scared/pushed away”.
According to the German etymological source that I am using (dwds.de), the words to shoot and schießen also belong to this family, which would make sense, as they are in a way a very quick shoving.
Don’t think of a gun – think of an old hunter gatherer propelling a spear.
By the same token, to scoot would then also be related.