The alternative, which the author of the linked article mentions, but doesn't really emphasize, is to report a best estimate of whatever effect you are trying to measure, along with some measure of uncertainty in that estimate.
For example, instead of "we failed to find significant evidence that right turn on red increases the expected number of fatalities," you say "our best estimate of the expected increase in fatalities due to right turn on red is 200 +/- 210."
This approach puts the most relevant information front and center, and it seems to me, encourages better intuitive reasoning. It's what engineers and most of the hard sciences do most of the time.
You do also need to say something about the meaning of your uncertainty estimate (e.g. it's 1 sigma, or 2 sigma, or 95%), or alternatively, there needs to be an understood convention for your field.