The replication crisis is happening throughout science, including neuroscience, genetics, and oncology, to name a few:
http://www.nature.com/nrn/journal/v18/n2/abs/nrn.2016.167.ht...
http://onlinelibrary.wiley.com/doi/10.1111/j.1360-0443.2009....
http://www.slate.com/articles/health_and_science/future_tens...
In fact, some analyses suggest replication might be worse in the neurosciences than in psychology. The neurosciences are plagued by many of the same conceptual ambiguities in psychology, but with even smaller, expensive studies that are underpowered with more incentives to avoid sunk cost problems.
The most accurate summary is that replicability problems affect the biomedical sciences the most in general (empirically speaking, psychology tends to be classified as a biomedical science, in topic analysis studies and so forth). Follow the money. You have lots of competition for limited resources, lots of money on the line, and soft or absent tenure protections.
The link you provide amounts to nothing more than ignorant intellectual bigotry and killing the messenger.
It's ironically sad, in fact, because psychologists and behavioral scientists are the ones exposing the replication crisis and doing the most to try to address the problem.
The type of rhetoric in the linked essay amounts to a whitewashing of replicability problems in other areas, by denying the problem exists and is only true of "lesser sciences." The sort of defensive response represented by the linked essay allows scientists in other areas to avoid accountability and the threatening implications it has for other fields.
Levels of analysis problems happen in all sorts of intellectual endeavors. Arguing that psychology will give way to the neurosciences is akin to arguing that nothing in the computer sciences is a legitimate topic of study unless it involves bare metal issues.
Having said that, I also distrust this paper.