I imagine that this is the same limitation that we have with the Steam Link only supporting 4 controllers since it relies on xinput to stream controller input and xinput has a 4 controller limit.
As far as I know, this is a limitation of Steam's input streaming service.
However, we haven't updated our controller library (Rewired) in several months and it's POSSIBLE that they've implemented some sort of workaround. That's something I've been meaning to test, so this gives me another reason to do so.
In the meantime, you can still play 5-10 players with Online mode (unfortunately this would require each player to have their own copy). You could also try getting weird with a hybrid approach. Have 2 copies of the game with remote play together, go to Online and have up to 3 additional players join each copy by pressing A. Your lobby would look like accountA, accountA, accountA, accountA, accountB, accountB, accountB, accou...