Hello, a couple of quick comments, first: Welcome to Eugene! Second, Realtors are more focused on Selling homes that finding rentals, so hopefully you will get some property managers to respond and help you find a nice rental. BTW, this Brittany location is nice. Third, the State of Oregon rules for real estate agents does not allow us to comment on schools, they want that reserved for parents to research. So, if you do get an agent to discuss schools, it might be because they have kids in those schools.
If no property managers respond, which don't expect, I suggest that you Google Eugene rentals or something like that and you will have plenty of hits to work on.
Again, welcome and enjoy,
DiMar Real Estate