236 is a massieve sample size and it is more than sufficient for showing an effect here. The p-values are below 0.001, and while p-values aren't everything, it's a strong indicator that the result is statistically significant.

Sure, the correlation is statistically significant. Without a control to infer causality the results become much less interesting.

To infer causality in an epidemiological study such as this you need very large cohorts with very specific results.

Even with huge cohorts you still have relatively indefensible results. The 7 country study is a good example: https://en.wikipedia.org/wiki/Seven_Countries_Study

Particularly since one of the connections they postulate is between antibiotics and asthma, I'm curious how they control for respiratory infections which seems like a much more likely common cause.

p-value is almost irrelevant if you don't know how many overall statistical tests were performed. Finding a p<0.001 "significant" result after testing 1,000 different variables is far less meaningful than finding one after testing 5 variables.