"then repeat a time or three, then give a good rinsing."
And then you beat them against a rock in the stream?
The shop I worked at many moons ago always washed their rags (sent out to a service), so that's what I follow. Very convenient to have my own machine now!
I do toss the really greasy ones, like if used for removing black goo from u-joints, etc. But I try to use the already near-death rags for that. And buy a new pack of rags maybe every five years.
For the red rags in the washer, filling it (loosely dropping in) only about 3/4 full worked well. But if very dirty I'd only fill half-way. And I've always run the full heavy cycle twice when doing these rags.
P.S. Greasy rags can auto-ignite, so be sure to store them in a well ventilated, non-combustible, cool-location can. I always let the wet (with water or solvent, etc.) rags dry/air out before putting them in the to-wash container.