Often it is impractical to perform large studies. Partially by logistics and often by funding. But if there are lots of smaller studies you can aggregate the data to check for overall significant results. But mainly they are helpful to give a "survey" of the current research instead of having to link to 10 different studies and hope someone else sorts out what the data says.
But as you point out, there are two major flaws in the assumptions. First, that the scientific procedure is sound. Secondly, that the data is handled properly, and thus you can take the summary and back out the underlying data.
Unfortunately trying to fix the first is really, really hard.
The second is somewhat mitigatable. As the Nature article suggests, you could publish the underlying data (anonymized of course). This would help in two ways. First, the meta-analysis could check for confounding variables to control across all of the data. The second major one is it would help people spot fraudulent data.
However, as anyone who handles datasets knows, publishing and wrangling data into a useable state from multiple sources is a serious pain in the neck. Plus a lot of concerns about how de-anonymized the data would be. As we've known, with enough metadata it can be used to identify individuals. And publicly publishing dais data would definitely allow for some serious sleuthing work to be done.