I do research on the topic of the article. I actually initially clicked on it because the title seems so uncannily similar in content to an article I wrote that I thought they might mention something about it. They did not but it's still really salient to me.
The article is sort of strange in that it brings up these issues and then fails to discuss the huge range of research in this area. The p-factor stuff is sort of hot lately but is just one subtopic in a vast area, somewhat controversial, and more of theoretical interest than clinical utility.
Part of the problem is that there are different purposes for classification systems. I think you're right in that categorical diagnoses have all the appeals you mention at least among some subgroups, but those are a little detached from (1) how many clinicians actually think about a lot of problems, and (2) how many researchers think about these things. Putting aside the "syndrome"-favoring clinicians, a lot of clinicians focus more on behavioral patterns, which tend to be very specific, more targetable, and far removed from broad categorical labels. Many researchers are moving away from these categorical labels also, for many reasons, but mostly generally because the patterns you see in behavior don't map onto the categories in the DSM in reality. Think blood pressure or height as analogies for depression or disorganized thinking, rather than Huntington's disease or malaria.
RDoC is important to mention but it's ending up to be about as controversial as the DSM for various reasons. First, it suffers from the same "names by committee fiat" as the DSM, which is problematic when you are dealing with something like neurobehavioral pathology, which can be examined at many levels and from many directions, which is fuzzy and complex, and about which we know very little. Second, despite its noble intents, was kind of developed by neuroscientists with little connection to actual clinical human behavior. I don't mean any offense to them, but when you get together a bunch of researchers, a substantial number of whom study rodent neurobiology, you're going to start losing touch with what clinicians are actually wrestling with in clinical settings. Look for psychosis or subdimensions of psychosis, for example, and it's missing from RDoC. I like to think of RDoC as "revenge of the neuroscientists," people who got tired of trying to connect their research to human public health, and decided to just redefine problems so it's closer to their own research. A similar initiative is HiTOP, which is more phenotypically focused, and uses statistical/quantitative models for classification rather than committee decisions (I'm not saying HiTOP is better, it just is a similar but different approach from RDoC).
I think the issue is not whether or not mental disorders should have names, it's whether or not there are mental disorders per se, or how to best describe mental-neural-behavioral patterns. You need to be able to quantify to study something scientifically, and quantifying implies labeling; the question is how.