What is the exact model of that machine? Does it not have any FR mode available? It may be one of the later-generation LG clones, and not one of the prized JVC models often talked about on this site and some others. Then again, if it cleans that much of the image, it can't be an LG (unless it's an LG based on LSI chipsets).
The JVC VCR should clean up the image, yep -- that's very typical.
For those tapes where the TBC caused issues, try it without stabilizer -- basically turn off everything. Leave the static DNR in NORM or AUTO mode, not EDIT, SHARP or SOFT -- and that's it. See how it plays. Sometimes the stabilizer can cause image jitter (vertical jumping), so be careful with it. Monitor the video carefully.
Sometimes those overpriced cables are worse than the cheap add-ins that came with the decks. I don't much care for Monster s-video cables, preferring the Philips cables from Lowe's or
Walmart.