Hey guys, first-time poster, here.
Respecting the fact that this is a Line6 forum, I hesitated to post because I am a very contented GNX4 user for the past 12 years. I now have SEVEN of these units. Gigging without a backup is not an option for me. I only have so many because they can be had for next-to-nothing in as-new condition.
Clearly, the OP has a fundamental setup issue that is likely not at all related to a limitation of his modelers. The fact that one of the harshest-sounding presets, "Satch", is the one that sounds good to him, speaks volumes.
The following links, transformed what I initially regarded as a cheesy toy, in to something that I now regard as an extension of my guitar and, I suspect that the advice can be of use with all modelers:
http://mfxsupermodels.com/MFXNEW806/CUSTOMIZE.html
http://mfxsupermodels.com/MFXNEW806/EFFECTS.html
http://mfxsupermodels.com/MFXNEW806/MFX_SS_LIVE.html
http://mfxsupermodels.com/MFXNEW806/ASSIGNS.html
"MY AMP SOUNDS ARE DIFFERENT THAN THE HEADPHONE SOUND..WHY IS THIS??"
http://mfxsupermodels.com/MFXNEW806/FAQS.html