Why is Principled Hair BSDF not showing the underlying texture paint?

Thank you for the detailed response, @Hunkadoodle. I was able to set up the materials as you suggested, and while – theoretically – I have the Principled Hair BSDF working with the texture map supplying the colors for the fur, the particular texture map that I’m using (with a lot of tan/brown colors) doesn’t want to cooperate with the shader in any meaningful way.

What I mean is, I can take a texture with bright/vivid/saturated colors and put it behind the Principled Hair BSDF and can see the results just fine. For example, here’s a pure red texture:

But for more muted colors, the colors don’t really appear to come through very well. Depending on the Roughness/Radial Roughness settings on the Principled Hair BSDF shader, the range of colors appear to be anywhere from silvery black to white…maybe with a little bit of tan thrown in.

Here are some examples that also show the Principled Hair BSDF Roughness and Radial Roughness parameters:

As a point of reference, here are what the colors are supposed to look like:

Still, though, just sending in the intended texture but with a straight Principled BSDF shader appears to give the best results visually:

Part of me wonders if what I’m seeing is the result of using a HDRI for the background – it appears that the blue from the HDRI sky is affecting the Principled Hair BSDF colors pretty strongly. Maybe it’s physically accurate, but it just doesn’t look good…at least, not with the settings I’m using.

According to this post, I guess at least one Blender user doesn’t think Principled Hair BSDF is very good for fur…I’m leaning in that direction too, unless I can find a combination of parameters that makes the fur colors look the way they should.