I use them a lot to control the tone of what I write. Since periods can indicate a serious or angry tone[1], I often need some other way to indicate a separation between two complete sentences. For example: "It looks like you forgot to do X; that's needed to do Y." Using a period could be perceived as anger, but using a comma would make that sentence a comma splice. [1] https://youtu.be/fS4X1JfX6_Q?t=3m3s
I use semi-colons as a sort of conjunction, not unlike the example you gave, in part to give me a way to lengthen sentences that would otherwise seem much too short.