The “with enough data, the numbers speak for themselves” statement has several meanings.
In one sense, if you can observe real phenomena, you don't have to guess at what is happening. For businesses that collect troves of it, they may need statistics 'less' because the sample size may approach the population size.
But calculating basic (mean, standard deviation, etc.) statistics is hardly the most interesting part. Inferential statistics is often more useful: how does one variable affect another?
As the article points out, the "... the numbers speak for themselves” statement may also be interpreted as "traditional statistical methods (which you might call theory-driven) are less important as you get more data". I don't want to wade in the theory-driven vs. exploratory argument, because I think they both have their places. Both are important, and anyone who says that only one is important is half blind.
Here is my main point: data -- in the senses that many people care about; e.g. prediction, intuition, or causation -- does not speak for itself. The difficult task of thinking and reasoning about data is, by definition, driven by both the data and the reasoning. So I'm a big proponent of (1) making your model clear and (2) sharing your model along with your interpretations. (This is analogous to sharing your logic when you make a conclusion; hardly a controversial claim.)