How to interpret multiple hypothesis tests where $p$-values > 0.05 but test statistic signs are all (or nearly all) negative (or positive)? This situation often arises when I'm running tests on healthcare data for my job.
Taken individually, the hypothesis test $p$-values seem to show no statistically significant relationship between $X_1$, $X_2$, . . . , $X_k$ and $Y$.
However, taken as a family of tests there is a clear relationship between $X_1$, $X_2$, . . . , $X_k$ and $Y$, indicating the variables are not independent.
Should I be running a "meta" hypothesis test on the results of the initial $k$ hypothesis tests?

The main point is that you cannot reframe your hypothesis based on the data you have already observed. The results will never generalize to another sample.
The "sign" of the trend for each hypothesis shouldn't matter theoretically. What we care about is the correlation between tests; in the case that tests are highly correlated, we know a Bonferroni correction would be conservative. Effects that are of opposite signs in a sample can come from probability models where the tests are highly positively correlated, or in fact any scenario can be dreamt up here.
But, alas, you didn't apply a Bonferroni correction! You would need to compare p-values to the $0.05/k$ alpha level! There is basically nothing you could have done to conserve the familywise error rate (FWER) and find a significant result. The FWER is a well defined operating characteristic of multiple testing. When you refer to "tak[ing] a family of tests", testing each hypothesis at the overall $\alpha$ level is already an anti-conservative approach - the actual false positive error rate is higher than stated, i.e. it is statistical cheating, or "p-hacking".
Based on this, you should report the results as-is and be done with it!
