I don't think you even need to be pushing people to do Bayesian stats. You just need to force them to graph their data properly. In *a lot* of biological and social science sub-fields it's standard practice to show your raw data only in the form of a table and the results of stats tests only in the form of a table. They aren't used to looking at graphs and raw data.
You can hide a lot of terrible stuff that way, like weird outliers. Things would likely improve immediately in these fields if they banned tables and forced researchers to produce box plots (ideally with overlaid jittered raw data), histograms, overlaid 95% confidence intervals corresponding to their stats tests, etc, etc.
Having seen some of these people work, it's clear that many of them never make these plots in the first place. All they do is look at lists of numbers in summary tables. They have no clue in the first place what their data really look like, and know good knowledge of how to properly analyse data and make graphs. Before they even teach stats to undergrads they should be making them learn to plot data and read graphs. It's obvious most of them can't even do that.