I don't think so, I can't see where any larger than #10 would be *required* under article 630, can you show me if I am missing something? I agree all assumptions are for copper wire.
Skin effect is not appreciable at 60 Hz with wires this small.
To the receptacle, #6 is required since multiple loads could/might be plugged into the receptacle. When you are dragging 40' of cord, #8 is needed for the length. The #10 pigtail on the welder is a short either 4 or 6 feet depending on the year of the welder and the code cycle. The welder is considered an appliance and manufacturers get away with having the appliance pigtail be undersized and kept short to save money. If you are running the welder full bore and over the duty cycle, (we all do) you can both burn up the cord and/or damage the welder. Electrical service calls to fab shops are all about undersized and over length cords.
You are correct regarding the skin effect on small conductors, of course, building wire is manufactured stranded #8 and larger only. Solid #6 can be found only in bare for grounding electrode installation. At work, we quit pulling solid wire for anything in the mid 70s due to excessive labor for pulling and terminating solid wire.
The only other suggestion I might offer, is to review the jacketing on the cord you buy. some of the premade cords are made for generators with lightweight jackets. Look for an SJOW cord jacket (I included the letter designations fyi.) An SJOW can be driven over with no lasting damage, it can have heavy objects dropped on it and other forms of abuse not planned for but usually happening in a shop.
I also included the designation "P", zip cord is what is used for your welder pigtail.
S = Service Grade (also means extra hard service when not followed by J, V, or P)
J = Hard Service
P = Parallel cord (also known as zip cord)—Always light duty
O = Oil Resistant*
W = Outdoor—includes sunlight resistant jacket and wet location rated conductors (formerly "W-A")