I'm in a debate with a coworker and I'm starting to wonder if I'm wrong but the internet is confusing me more.
We have continuous data $[0, \infty)$ that is retrospectively selected on individuals. The selection is non random. Our sample sizes are $\approx 1000$. Our data is heavily skewed towards the left with some strong bumps towards the tail.
My strategy is to look at the distribution of the data before statistical tests between two groups via histograms, q-q plots, and Shapiro Wilk test. If the data is approximately normal I use an appropriate test (t-test, ANOVA, Linear Regression etc). If not I use an appropriate non-parametric method (Mann-Whitney Test, Kruskal-Wallis, Bootstrap regression model).
My coworker doesn't look at the distribution if the sample size is >30 or >50 he automatically assumes it is normal and cites the central limit theorem for using the t-test or ANOVA.
They cite this paper: t-tests, non-parametric tests, and large studies—a paradox of statistical practice? and say that I'm over-using non parametric tests. My understanding is my method would tell me if it's appropriate to do a normal distribution though because I thought that for heavy skewed data the n to reach ~normal distribution was higher. I know given a large enough sample size it would eventually get there but especially for the smaller sample sizes isn't it better to check? To me it makes sense that since multiple tests show that the data isn't normal it's inappropriate to use normal distribution then. Also if needing a sample size of 30 was all you needed for assuming normality why is so much work done on other distributions in statistical software? Everything would be normal distribution or non parametric then. Why bother with binomial distributions or gamma distributions? However they keep sending me papers about central limit theorem and now I'm not so sure. Maybe I am wrong and I shouldn't bother checking these assumptions.
Who is right and why?