I think your initial process for trying to analyse this colour palette was inherently flawed since it seems to assume that selecting and using colour schemes is generally model based. As others have noted it certainly doesn't have to be and the selection process can often seem entirely random. Without knowing anything of the project or audience, not to mention the designer, it's nearly impossible to explain the series of choices that led to this result. But it is possible to try to explain why and how the combination works.
The palette works, but because it DOES NOT harmonise.
These four colours obviously don't fit neatly into any pre-defined colour model. Consider the same colours used in a gradient; would you still say they worked well together? Probably not.
So we can determine that this lack of complete harmony creates a certain amount of visual tension because there is a marked level of difference between them. One of the basic principles of design is that the careful use of differences creates visual interest. I would conclude from this that a choice of hues like these is in order to create several distinct areas of interest within a confined space, whether website or on a printed page.
Not just hue...
It's also important to note that there is not a terribly high level of contrast between the colours. This can be seen from the Brightness and Saturation values in your screenshot and the place they occupy on the colour wheel. You could also consider the 4 shades reduced to greyscale.
Luminance variety can be terribly important and this article does an excellent job of explaining why. But for the example at hand I would guess that the use of three tones with similar Brightness and Saturation values is in order to dial back on the visual tension between the shades slightly. If you are using several colours in creating separate areas of interest then difference in Hue is fine, but high levels of contrast will create a visual mess. In any situation where chosen Hues are largely different this matching of Saturation and Brightness is a neat way of promoting a certain amount of harmony. For example...
An interesting side note is that the use of muted tones has been said to be good for concentration. In the book Color Psychology And Color Therapy the colour researcher Faber Birren discusses how softer and deeper colours promote mental and visual tasks better. But I digress.
I would reason that the green shade being positioned at the top in the given example is no accident. It is the most distinctly different from the others, as you can see from the colour wheel, and so it also exerts an influence on the other shades. Colour psychology tells us that in western cultures green has connotations like growth, health, nature and communication. Is there supposed to be a link to any of those in this case? Without knowing where this example is from I obviously can't say, but if I was going to bet I would guess that there is. Defining a palette from a single dominant shade generally means that the overall mood is set.
Finally, I have to point out that in my experience the selection of colours is almost never random. Building up a palette might start with the dominant colour from a logo, or at the behest of a client, or to create a certain mood or tone. But like a Michelin stared chef creating a new dish, or a jazz musician playing a solo, what might seem to be erratic more often comes from reasoned choice.