I know that I'm beating a dead horse, but:
Since they were mentioned earlier, here's a link to an old thread about eyelet-doublets:
http://myArmoury.com/talk/viewtopic.php?p=222875
Some of the linked photos show what it looked like - lots and lots of very small rings/eyes, very closely joined together (one photo shows part of the doublet and some mail nearby, and the eyelets are much smaller than the mail rings). I don't know why this would have been used in favour of a coat of plates or brig, but I think that the combined effects of quilting several layers of cloth and the fact that the rings are so small and close together is what made it viable as armour.
However, most "
ringmail reconstructions" (like the original poster's attached picture) use much larger rings than the eyelet doublet, with bigger gaps - if an arrow or strong spear thrust could sometimes break a ring or two in real mail and pierce a hauberk, the same weapons would make mincemeat out of someone wearing armour composed of similar rings stitched independently to a cloth backing with little or no overlap, with its relatively large unprotected gaps between rings. In short I don't think comparing it to eyelet doublets makes sense.
The cost arguement also doesn't make any sense on closer inspection. Whatever the reduced cost associated with using a smaller number of rings than for proper mail, it would still have been easier and much quicker to hammer out small metal scales and stitch them onto armour - and the resulting scale armour would offer considerably better protection than "ring mail" because the plates are solid and overlap. We know that for some reason the west and northern europeans
didn't use scale armour in the viking-era, but it
was widely used by other cultures and in other time periods, and the vikings of the varangian guard might have at least seen it. Other than eyelet doublets, the only historical example of anything even loosely similar to "ring mail" armour is a single obscure garment from somewhere in Asia.
So why would anyone bother to make a more expensive, time-consuming and ineffective version of scale armour?