@Baktrian, unfortunately the asset you are using is not set up in a way we would expect for any of our shaders. Looking at the base glb, there is only one texture - the base color texture - defined in the glTF, and the other parameters are defined as factors in the file:
When loading the model as is, the render looks like a decent start to the render from sketchfab:
However, it is missing its emissive texture and the roughness variation you see in the original on sketchfab. However, when I toggle through the material channels on sketchfab, and then look at the textures you supplied for the model, I see the issues you are having. The texture you are passing as a specular texture is not, in fact, a specular texture but a channel packed texture containing several textures, one per channel. You can see here that the red channel of your specular texture contains what would be your smoothness texture. Normally, you would assume author metallic roughness, but engines like Unity require metallic smoothness, so the texture is the inversion of a roughness texture like we see in sketchfab.
Which is an inversion of the roughness texture found in sketchfab:
The green and blue channels contain different information entirely, which would likely drive a custom shader. Arguably, the green channel contains a metallic texture, which you can see when overlaying the green channel with the red. The fully green bits fall on the metallic parts of the model.
From what I see in Sketchfab, it’s not using a metallic texture at all, so there seems to be data in the textures included with this model that aren’t referenced for the original render. My best suggestion would be to peek into each texture and see what is there and what each channel could potentially represent. Without seeing a full render of the character using the correct shader, it’s hard to guess what each channel is used for. The blue channel of that texture has detail in it that I don’t see in any of these renders, so I can only guess at what the shader would do with it.
This could be a specular F0 texture for a metallic roughness material. Or it could be something else entirely. I doubt it’s a glossiness map to throttle specularity, unless it’s an inverted one as the values don’t make sense for glossiness.
If you can break down what each channel is doing, you can reconstruct the textures so the channels contain data that our shaders expect, or you can go the custom shader route. I prefer working with node materials, and this one seems like a good candidate for that. I hope this helps point you in a good direction, but feel free to ping back with more questions.