Some laundromats have a washer dedicated for greasy clothes. It might be worth checking out.

But honestly, I hate the idea of washing all that stuff down the drain. I think it's actually more environmentally responsible to send paper towels to the landfill where at least there's supposed to be an impermeable liner to keep it from getting into the water.

I use rags to wash my frame, they do get greasy, and I confess to using Chicken Little's method on them. But for the drivetrain itself, with that amount of chain lube on it, I use disposable towels.