While I agree with the article's headline/conclusion - They aren't innocent of playing games themselves:
Take their sentence: "meeting online nudged the divorce rate from 7.67% down to 5.96%, and barely budged happiness from 5.48 to 5.64 on a 7-point scale" ... Isn't that intentionally misleading? Sure, 0.16 points doesn't sound like much... but it's on a seven point scale. If we change that to a 3 point scale it's only 0.06 points! Amazingly small! ... but wait, if I change that to a 900,000 point scale, well, then that's a whole 20,571 points difference. HUGE NUMBERS!
But I think they missed a really important point - SPSS (one of the very popular data analysis packages) offers you a huge range of correlation tests, and you are _supposed_ to choose to best match the data. Each has their own assumptions, and will only provide the correct 'p' value if the data matches those assumptions.
For example, Many of the tests require that the data follow a bell-shaped curve, and you are supposed to first test your data to ensure that it is normally distributed before using any of the correlation tests that assume normally distributed data. If you don't, you risk over-stating the correlation.
If you have data from a likert scale, you should treat it as ordinal (ranked) data, not numerical (ie. the difference between "Totally Disagree" and "somewhat disagree" should not be assumed to be the same as the difference between "somewhat disagree" and " totally agree") - however, if you aren't getting to the magic p0.5 treating it as ordinal data, you can usually get it over the line by treating it as numerical data and running a different correlation test.
Lecturers are measured on how many papers they publish, most peer reviewers don't know the subtle differences between these tests, so as long as they see 'SPSS said p0.5' and they don't disagree with any of the content of your paper, yay, you get published.
Finally, many of the tests have a minimum sample size that should ever be analysed. If you only have a study of 300 people, there's a whole range of popular correlation tests that you are not supposed to use. But you do, because SPSS makes it easy, because it gets better results, because you forgot what the minimum size was and can't be arsed looking it up (if it's a real problem the reviewers will point it out).
(Evidence to support these statements can be found in the "Survey Researcher's SPSS Cookbook" by Mark Manning and Don Munro. Obviously, it doesn't go into how you can choose an incorrect test to 'hack the p value', to prove that I recommend you download a copy of SPSS and take a short-term position as a lecturer's assistant)