One minor (or, what could be a major one) point. The main purpose of the separate ground wire is to provide a low resistance path so that a fault between a hot wire & the metal of your trailer will cause the circuit breaker or fuse protecting the circuit to trip. If the fault current is not high enough to trip the supply protection, even with your own "earth" connection, you can receive a shock between the fault & the earth.
Adding your own connection to the earth will often not cause a fuse or circuit breaker to open during a fault because the resistance of the earth or your connection to it is too high to carry enough current to cause a trip. For example, in our area you are required to drive 2 8' ground rods spaced 10' apart to provide a low enough resistance earth connection to satisfy our local utility.
If the campground ground connection is bad, making your own provides a false sense of security unless you are sure the resistance of your earth connection is as low as what the campground wiring would have provided.