Alright I'll try this once more. The P-value in these studies is an attempt to show a statistically significant difference between the groups investigated. Larger p-values mean there are not statistically significant differences for that value. There is no way that the statement "these two groups have similar depression levels" could be proven by a low p-value because a low p-value would show a difference between the two groups. The fact that the two groups had high p-values for the measured value of depression is the reason they conclude the two groups have the same level of depression. If the p-value had been low then they would conclude that the two groups are different. There is no way to conclude the two groups are the same with a low p-value, it's just not how the analysis works.
That is my reading on how the studies are designed, if you have more information on this than I I'd be interested in hearing it, but simply repeating the idea that only conclusions with low p-values are valid is meaningless. p-value is not "the error in the conclusions drawn", but a means of showing whether a null hypothesis has been disproven. If the null hypothesis is that the two groups are the same, and your p-value is high this means that the two groups are not different to a statistically significant degree. This is what the study stated.
So, given my reading of the study, what are your credentials to be discrediting the analysis of experts in their field? Because I'm pretty sure these guys understand P-values much better than you or I.
Here's some info to back up my interpretation:
You're right in that a high p-value validates the null hypothesis. But the value in that study isn't high enough to validate the null hypothesis either. Just do 1-P and you are back to proving how worthless the study is. In your case, you'd have to prove p>=0.95 to validate the claims.
That doesn't really make sense. To restate what a p-value indicates: P values indicate the odds of seeing the results given the alternate hypothesis being true. A large p-value would require an almost entirely uniform sample and would not exist in circumstances where you have natural variability. As such, the "1-p" calculation is not really something that is done except perhaps in some fringe scenarios where you are trying to prove some uniformity. Further, you talk about "validating the null hypothesis". That also doesn't really make sense, as the null hypothesis exists as a rejection of the alternate hypothesis. To try to assert that something is statistically significantly insignificant is kind of nonsense.
To quote this study: https://www.ncbi.nlm.nih.gov/pubmed/15080563
"In randomized controlled trials, main endpoint p-values larger than p=0.95 will be rare, because they would indicate similarities closer than compatible with a normal distribution of random data samples."