My parents live on the Yampa b/t Milner and Hayden and I have floated the mile or so that runs through the property tons, but I've never heard of anyone rafting from steamboat to deerlodge. I hope you like flat water. a lot.
You are right, it is pretty much all private property, but there are a few spots of BLM and public land. I don't think those spots allow camping, but i could be wrong. The one that pops into my mind first is b/t hayden and Craig. It's a good 30 miles from steamboat and it's the only public land I know of to that point. Downstream from that I think there is some just before Maybell.
Then of course the put in for cross mountain is great camping but they charge a fee. 70 miles of flatwater should be a good warm up for cross.
I personally think you are nuts, but the world would be a boring place if there were no nutjobs out there, so have fun.