Intraclass correlation with skewed dataset

Background: There is data on 20 people taking the same test 5 times each to look at the repeatability of the test.

I understand that intraclass correlation is usually the best test to use to look at a question like this. However, the 100 data points are not normally distributed (clearly evident by a large left skew on histogram) and also do not pass a Shapiro Wilk's test for normality. I have tried some transformations of the data such as log, cube, sqrt, box-cox and none are able to sufficiently correct for the non-uniformity. However, the averages of each participant are normally distributed.

Based on this, I have 2 questions. I have read that once a dataset is sufficiently large (>40), non-normal distributions do not affect parametric tests like ICC very much. Could this be assumed in this case (since the number of subjects is below this, but the number of data points is much larger than 40)? If not, what test(s) would you recommend looking into to specifically access how repeatable the test is overall?