Flight plans are supposed to include your mobile phone number and after calling your filed destination aerodrome, this will be their next attempt. If I file via RocketRoute, the plan also includes RocketRoute's phone number.
Regarding closing flight plans in the air, this doesn't always work depending on the country. It used to be quite relaxed in Germany but nowadays they refuse unless you are in the traffic pattern, at which time you should be talking on another frequency. I've heard it many times on the radio that FIS refused closing a flight plan because they considered the aircraft to be too far away from its destination.
I haven't heard lately about pilots getting a bill for SAR because they forgot to close the flight plan. They have so many options these days like the mobile phone or calling the airforce which operates a country wide 3D RADAR system that is much more capable than the civilian system. What happens a lot more often these days is SAR charges because of an ELT gone wild.