I think we might be bumping into confusion over different stranding types found in wires, and possibly lugs being used outside of their proper listing.
As already mentioned, there are different classes of stranding. You can have the minimal stranding used to make large conductors flexible enough to be installed without using bending machines, and you can have the fine stranding used to make conductors intended to be continuously flexed such as welding cables.
Very often lugs intended for minimally stranded conductors are not suitable for fine stranding. Ideally the instructions would specify the permitted stranding, but this is probably buried in the listing requirements.
Lugs for fine stranded wire are usually compression type, not set screw type.
I've specifically seen copper foil or ferrules required for mechanical connections to fine stranded wire on some connectors, mostly 'cam-lock' type connectors. Not on lugs that I recall.
But I have encountered lugs where I've used normal stranded wire, at the small size of the allowed wire range for the lug, tightened the screw, and found that the screw splays the strands apart and several strands end up between the screw and the body of the lug.
Jon