Example: you believe a newly found plant species is toxic. You give it to 17 "grad students volunteers", while giving a placebo to 17 others. All in the first group die aa gruesome death within 20 hours. None of the others do.
Result: yes significance. (also: tenure!)
I'm not saying that this study is significant (the statistics seem to be slightly beyond my event horizon), and your criticism also stops short of an outright dismissal of the research. But sample size alone makes for a bad measure of quality. Yes, even p-values are better.
Effect size is very important in this. To continue your grad student murder example, it's completely trivial to determine which plant a student was given, based on whether they are dead or not. It becomes trickier if you measured something a bit less cut-and-dry, such as the incidence of headaches, or variance in a few voxels of a noisy MRI.
Looking at it either from a machine learning or statistical point of view, using such a small sample is problematic.
This is the chronic issue with fMRI studies, since administering an fMRI is extremely expensive, and has led to some very difficult to reproduce results in the field.