The LED on the EZX10RF blinks when it is receiving an X10 RF signal and writing that X10 message on the powerline or writing the Insteon On/Off Group command sequence for the Insteon device that has been linked to the specific X10 code. The LED also blinks when it detects Insteon or X10 traffic on the powerline originating from another device. This is the less likely possibility in your case. It would be apparent if the LED was blinking when you pressed the ICON paddle.
Are you using X10 or Insteon messages to actually turn On/Off the ICON switch? If X10 the two X10 messages are sent from the EZX10RF in the blind with no expectation of a response from the ICON switch. If the ICON switch was linked to the EZX10RF when the X10 code was established from the EagleEye, then the EZX10RF will retry the Insteon command to the ICON switch several times until it receives an ACK from the ICON switch or the max retry count is exceeded.
You did not mention any cross phase coupling device(s) when you described your configuration. What that should be depends on whether you are using X10 or Insteon messages to control the Insteon devices. The ICON wall switch could be on the opposite 120v leg to where the EZX10RF device is plugged in making powerline communication intermittent. Without explicit phase coupling you are dependent on a 240v appliance to provide that coupling. An electric hot water heater, an electric dryer, etc. to get the powerline signals from one 120v leg to the other. Works okay when the appliance is on, fails altogether or is intermittent when the appliance is not on. As a test you can try plugging the EZX10RF into a circuit that you know is on the same 120v leg as the ICON switch.
Don’t know about plugging an Insteon device into a GFCI protected circuit. I would worry about that only if that is the only device you are having problems controlling.
The two primary questions, are you using X10 or Insteon messages from the EZX10RF to control the ICON wall switch; what if any cross phase coupling do you use?