Nope. It's "If we did this infinitely more times, there's a 5% of those samples wouldn't have significant results". It's a subtle but important distinction.
Though I'm surprised that his advice wasn't "Report confidence intervals at least". There's much more meaningful information in a point estimate and confidence interval than "p < 0.05"
While built off the same information, and it's possible to do an ad hoc significance test off it, confidence intervals tell you more about the spread of estimate. Especially if, as the author is suggesting, you're not even reporting the actual p-value, but just whether or not it's below a particular threshold.
