Is the Brodie landing system still in use somewhere?
I'm not aware of the Brodie landing system used anywhere now. The system was actually a response to the pressing problem of non-availability of escort carriers in required numbers, similar to the British CAM ships, that launched hurricanes on one-way missions. The only difference being that the aircraft could be retrieved here. 

The main advantage of the system is its ability to operate from (cargo) ships and LSTs, which became less important as the war progressed and more and more aircraft and escort carriers became available. There were some disadvantages too- it required certain skill from the pilot for landing (especially if installed in a jungle or on the top of a building). Also, a 600' cable between two 65' posts can hardly be called small (and as @ymb1 noted, it is hardly practicable for larger aircraft).

Helicopters serve the same purposes- operate from anywhere with much heavier payloads and in non-existent airstrips routinely. 

However, similar systems are used for retrieval of small UAVs. For example, the Boeing Insitu ScanEagle uses a similar system of retrieval (though the launching system is different). From uavglobal.com:


  ScanEagle needs no airfield for deployment. Instead, it is launched using a pneumatic launcher, patented by Insitu, known as the “SuperWedge” launcher. It is recovered using the “Skyhook” retrieval system, which uses a hook on the end of the wingtip to catch a rope hanging from a 30-to-50-foot (9.1 to 15 m) pole. 




Scan Eagle retieval; By United States Navy, Mass Communication Specialist 3rd Class Kenneth R. Hendrix - Navy NewsStand Photo ID: 071114-N-5758H-065Navy NewsStand Home, Public Domain, Link



There are patents for a number of similar systems (eg 1,2 and 3). Interestingly, Insitu calls its skyhook system:


  ... the first capture system to eliminate runways and nets 


Apparently, they didn't get the memo.