Here is my final result.
The result using the first node setup:
And the result using final principled setup:
Let me start of by saying that smoke simulation and rendering is not something I am very familiar with, so despite being able to replicate the material fairly well, I don't necessarily understand everything behind why it works the way it does. But I'll explain what I can.
A quick summary of what I did:
Left the default color as white, and used the colorramp as the input for the Emission Color.
Increased Emission Strength to the value you had, but then dialed it back some as it was too strong. I noticed that the two images aren't exaclty the same, and further reducing the Emission Strength could probably fix that.
Increased the Density to half of the value you used in the first setup, as this looked closest.
Left everything else as the default. I tried changing the color and temperature attributes, as well as the Blackbody settings, but they seemed to have no effect with this setup.
So, from what I understand at the end of this, the Principled Volume Shader is basically setup to replicate the original node setup you have, and it does it well. There does appear to be some differences in how the shader works compared to the old method, namely in how the density works. I noticed that (before emission was added, which made the details hard to see) the Principled smoke looked more dense than the manual smoke, but if I decreased the density it would get to the point of areas being nearly transparent before the main density looked similar. More details were visible in the principled smoke, and since in this setup the smoke was colliding with the domain, the principled setup had more defined boundaries, whereas the manual setup was much less defined. Perhaps with some experimentation, or a better understanding of how the shader works, as my knowledge is limited, a more similar result could be achieved.
The good news in all of this is that smoke is now much easier to setup, as the final setup I arrived at was just this:
Hope that helps!