Statistics are consistently used in linguistics
when working with a representative sample. Etymology is not generally thought of in this way, because individual words are historical accidents, not representative samples (there's just one data point), so we can't reliably generalize. Statistics
are used more generally to compare languages, e.g. to see if they are related (where the sample is many words, not just one), but even then it still is controversial and unclear to what degree such an approach (e.g. a "big data" method) works better than the traditional work "by hand" by experts looking at details of etymology and intuitive plausibility of connections, etc.
A p-value is not a
desirable thing in science, nor required for the "scientific method". It's just one tool available to scientists, which can be used when appropriate. By the way, a p-value should not be "between 1/300 and 1/17". It should have a precise value, which is either higher or lower than the arbitrarily-selected cutoff point. So you might select .05 as the cutoff, and if your p-value is less than that, you consider the result to be significant (i.e. unlikely to observe that distribution by chance), and you report simply "p-value less than .05". You can also include other kinds of statistics, such as a confidence interval (that's what "between 1/300 and 1/17" sounds like), but that's something else. Regardless, there has been a huge amount of discussion recently about the general
unreliability of p-values in research, i.e. researchers saying "this result is statistically significant", because although the numbers are objectively calculated, the methodology for gathering the data and deciding what it represents is not. Replicability is a neglected area of science, and in general a number of recent papers have found something along the lines of "most published conclusions are wrong", specifically in the case of experiments trying to make statistical generalizations.
In short, dismissing traditional insights into the field while hiding your methodology behind a shiny "p-value" isn't going to make it any more correct. It all boils down to the methodology, and whether your
assumptions and interpretations of the data are correct.
"Big data" approaches to historical linguistics run into this problem: the idea is that if you feed enough data into a computer, eventually the calculated result will be meaningful. But it turns out that the assumptions build into the models, and lack of respect for traditional understanding of how these things works, simply leads to unreliable results. Here's a great Youtube video of a lecture about that:
https://youtu.be/4jHsy4xeuoQInnovation in science is crucial. But only when it's actually an improvement, and generally after building on a solid foundation of existing methods.