OP here. I can ride in September because I am now retired teacher lady, but don't know how to change my user name!
I know thousands are riding the great rivers of Europe, because I rode the Rhine 3 years ago, and the Danube last week, and saw dozens every day on the Rhine, and dozens every day on the Danube (at least as far east as Budapest; we only saw a few a day after Budapest.) If I see dozens on any given day (mostly before high season), then there are 1000s riding each river every summer. I believe there aren't as many riding the Mississippi because there is no ACA route on the lower Mississippi. I know the Louis and Clark (bike) trail exists, but I hear very little about people riding it!
You've all given me food for thought, especially jamawani. I will ride the Katy trail some day, but I had in mind doing a very substantial part of the Mississippi, enough to see it grow and change. Free-flowing isn't essential, as I like watching the river traffic. But I do want to be able to get close enough to the river to see it a few times a day, perhaps take hotel rooms or campsites along the river. That looks like it will be difficult. Anyway, thanks for the answers.