Here is a helpful post about using barrier to effect different routes on successive runs.
https://community.esri.com/thread/63897
I am experimenting with creating a point barrier for each link used in a route and setting it's mileage to the original miles * 0.1 as an added cost. The points stack up as a link is used more than once, so the cost keeps increasing. This does yield a nice variety of routes. The same route can get picked more than once, but I can filter those out. I wish I knew how many of these routes to generate before I can feel confident I have found the 10 or 20 shortest routes.
Without knowing how the routing algorithms work, it seems clear that they can't and shouldn't find every route before they identify the chosen route. So my original question cannot be answered by routing software. I think that an exhaustive list of possible routes is something that sounds useful, but would really just be a very long list of no value.