If you want to test if its the skreem id suggest putting the key in the ignition.. Turn to crank, turn the key off but dont remove it.. Press lock or unlock on the key and see what it does. If the vehicld reacts.. Id say its the anttenna or module... If not its fine....hopefully.

Im basing this diagnosis procedure on when i made my push start ignition.. I removed the anntena ring, taped an extra key in it and hid it under the dash. The problem was that if the key is left around the ring.. It will ignore any keyless entry signals from it or any other remote after you've energised the ignition.. Even the outer door cylinders.

So its a guess that if your antenna ring is shot.. Then keyless entry will still work after a start attempt