`Compiler.LoadModule` assumes that when `ps.HadErrors == true` the error stream is not empty. However, when `SilentlyContinue` is specified as the error action, the non-terminating error is not kept in `ErrorOutputPipe` of the cmdlet and thus does not appear in `ps.Streams.Error`. So when `ps.HadErrors == true` while `ps.Streams.Error` is empty, it suggests it's OK to ignore the errors because they are explicitly suppressed with `SilentlyContinue` error action.
So in my opinion, the expected behavior of `"using module .\mod.psm1"` in this case should be successful as if `ps.HaddErrors` is false.
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