You're mixing up generally irreproducible results and a situation where a proportion of reported results are not replicated.
Modern psychology is very much a science, using the scientific method. However, due to the difficulty of studying it, the requirements for publishing a result are low enough that many of them turn out to be incorrect (not reproducible). That these results are eventually found to be incorrect is a validation of the scientific nature of psychology.
ANY subject involving a complex, difficult-to-study subject is going to have the same problem. Most fields like that, psychology, medical science, ecology, systems biology, etc., prefer allowing reasonable sized studies (usually less than a few million dollars) to be published, knowing that they may be incorrect, then replicating the interesting ones. Particle physics has tended to go the opposite way, where high profile results are not published until the confidence is very high, but those results also cost billions to achieve. Lots of less high profile results are, of course, held to lower standards.
The real lesson to take from the problem of replicability is not snide "psychology isn't a science" but rather that being published in a journal (hopefully) means that the study was done in a scientific way, but is no guarantee that it's true.