"it'll be run and other applications will behave (they'll be evicted in worst case)."
So, you are just trading one failure mode for another (if the system is overloaded). Misbehaving applications cause failure in either case. Also, having the scheduler reject work that it can't finish in time is STILL breakage of the system.
If I have two tasks, one real-time, one not, I set the real-time task to a higher priority and it gets all the CPU it wants, when it wants it - no breakage is possible from lower-priority tasks.
Where is the "killer app" for the deadline scheduler? People have been looking for it for years and no one seems to have found it. Maybe it doesn't exist.