The Test::Builder::skip documentation suggests to me that my usage is correct.

No, it doesn't. skip() is documented under:

Other Testing Methods

  These are methods which are used in the course of writing a test but
  are not themselves tests.
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It has nothing to do with control flow. It reports one test as skipped and then immediately returns. If you run a test afterward with an actual testing method, that's a new test.

Of course, you should only use Test::Builder directly if you're writing a test module. Don't use it directly to run tests. Use Test::Class and Test::More.