I'm struggling with Phantom 3 standard short range in 2.4G video transmission.

I found the U.FL antenna connectors on the 2.4G module in the remote controller are short-circuited.
When I tested circuit test with voltage meter connecting outer rim and inner pin, it beeps.I think it shouldn't.  Is my remote controller faulty or is it normal?

Are you testing the cable only or the connectors on the circuit board. Because if you are doing a continuity test on the circuit board, you are doing a very silly thing and yes they may well show a low resistance.

The Phantom 3 standard range is shorter then the other models. Your environment can influence the range a lot, in the opened area you get more then in the city full of skyscrapers  and other buildings.

Based on your pictures,  you are actually testing continuity for the whole circuit,  and yes,  you will get continuity.  That does not mean a dead short.  Try putting your meter in the OHMS position and measuring the cable instead of the circuit board.