What little hay I buy is for mulch. One seller I bought from found out what I was using it for and I ended up with some poorly tied bales which couldn't have weighed more than 10 lbs; I was carrying them around with 2 fingers. I bought it because I needed it but ate the cost; there's no way I was passing that on to the client for what it was.
I also never bought from him again. Weeds and a few sticks don't matter so much for erosion control. What does matter is that they have to stay together, and I'm not buying the junk bales which didn't tie right, or are only half the weight for whatever reason.
Rained on is OK as long as they aren't waterlogged and the strings aren't rotten; sometimes I have to carry them a ways, or tie several onto the ATV so they have to stay together. I like the poly rope for my purposes, even though it means I have to pick it all up afterwards.
Mulch hay doesn't cost much less, if at all; except for your herbicide cost, it takes the same amount of money to produce that as for good horse hay.