Testing a hypothesis against a pre-existing dataset is a valid thing to do, and it is also almost trivially simple (and completely free) for someone with a reasonable computational background. There are researchers who spend a decent portion of their careers performing these analyses. This is all well and good- we want people to spend time analyzing the highly complex data that modern science produces- but we run into problems with statistics.
Suppose an analyst can test a hundred hypotheses per month (this is probably a low estimate.) Each analysis (simplifying slightly!) ends with a significance test, returning a p-value indicating the likelihood that the hypothesis is false. If p literally hundred-to-one. But you see the problem; even if we assume that this researcher tests no valid hypotheses at all over the course of a year, we would expect them to send out one paper per month- and each of these papers would be entirely valid, with no methodological flaws for reviewers to complain about.
In reality, of course, researchers sometimes test true hypotheses, and the rate of true to false computational-analysis papers would depend on the ratio of "true hypotheses that analysis successfully catches" to "false hypothesis that squeak by under the p-value threshold" (i.e., the True Positive rate vs the False Positive rate.) It's hard to guess that this ratio would be, but if AAAS is calling things a "crisis," it's clearly lower than we would like.
But there's a further problem, since the obvious solution- lower the p-value threshold for publication- would lower both the False Positive rate and the True Positive rate. The p-value that gets assigned to the results of an analysis of a true hypothesis are limited by the statistical power (essentially, size and quality) of the dataset being looked at; lower the p-value threshold too much, and analysts simply won't be able to make a sufficiently convincing case for any given true hypothesis. It's not a given that there is a p-value threshold for which the True Positive/False Positive ratio is much better than it is now.
"More data!" is the other commonly proposed solution, since we can safely lower the p-value threshold if we have the data to back up true hypotheses. But even if we can up the experimental throughput so much that we can produce True Positives at p < 0.0001, that simply means that computational researchers can explore more complicated hypotheses, until they're testing thousands or millions of hypotheses per month- and then we have the same problem. In a race between "bench work" and "human creativity plus computer science," I know which I'd bet on.