> Semicolons start a new thought, they don't mark an aside that lets you return to the original line of thought.

Sure they do. They're perfect for a related tangent without abounding the greater scope topic being discussed.

> I would have used parentheses in both places, and semicolons don't work in either one:

Parentheses work no question and I would argue are far more appropriate in that example since it's a minor elaboration/clarification and not a tangent, indeed, semicolons would not be appropriate for that.