I spent another afternoon in with the veggies a few days ago, and I think I understand the secret of using hay for mulch. If you take compacted hay, like that pressed into a flake or a bale, and let it get wet, it will hold the water just like a sponge would. How quickly it rots probably depends on how much of it is stem and how much is leaf, but my grass hay is not rotting quickly, even the flakes I'm using for paths and borders which have soaked up and are holding so much water it runs out when I touch them. Grass shoots sprout from their edges and will root where they touch the ground. You can throw another flake on top of it to stop the sprouting, but eventually it will sprout too, if it gets wet.
On the other hand, if you take the same flake of hay and pull it apart into loose strands before it gets wet, you can pile the loose strands up 8" deep, at least, and no matter how many inches of rain fall on it and through it, it will stay drained and dry, and will not sprout. Only the thinnest layer touching the ground will rot, and if it's thickly enough covered to keep light from getting through, none of it will sprout, and no seeds underneath it will sprout either.
That's why it works as mulch. Compressed hay—bad. Loose hay—good.
And it's still early in the season, I probably have more to learn.