Its the PIC's call if they want a flight plan or not for internal flights, they have to have it for international.
If they close it before arrival I really can't see how this can be the problem of the ATCO if they then muff the approach/landing up and nobody comes looking for them.
To me asking to close it early is just the same as not bothering with one in the first place. If they cancelled their flight plan there wouldn't be a lot you could say apart from "flight plan canceled at xx:yy" but I can see where you are coming from, from a bum covering POV.