The current trend of saying that "cutting p-values off at a specific value is bad" makes me worry. Now you can argue that your p=0.06 result shouldn't be rejected when really we should probably be pushing for stricter standards rather than inching towards looser ones. It also destroys the nice interpretation of p-values above. P-values were literally made to be cut off - if you want to stop doing that, you need to show me a coherent philosophy of what to do instead.
What I do think is true is the problem you have where part A of the experiment suggests X so you test X more directly in part B with a weaker but more specific test and get p=0.06 and now you can't publish. That's a dumb cutoff, clearly a p=0.06 test is likely to shift our belief towards X so it does nothing but bolster part A. Typically papers do this several times and the marginal 'failure' of one step should not sink the entire ship. This is a case where a Bayesian analysis might be more useful as it can incorporate weak evidence.
But the problem I see often is not that p-values are misused but that they were junk in the first place. For example, the widely-used DESeq2 (as well as some competitors in RNA-seq differential expression analysis) will happily spit out p-values of 10^-100 for an experiment with only four replicates in each of two conditions! There is no way you can get that level of evidence from just four replicates, even if the values are 0,0,0,0 and 1e6,1e6,1e6,1e6. The assumption of normality is reasonable near the mean but gets increasingly inaccurate in the tail, which is exactly where you end up when you do things like sort 30,000 tests by their p-values. In fact taking a p-value cutoff is probably the only reasonable thing to do here - that way you'll ignore the fact that it's absurdly small and just treat it as "small enough".