You can always try avrdude (instructions at https://github.com/synthetos/TinyG/wiki/TinyG-Updating-Firmware ), it just won't help you. No flashing light == no bootloader.
But you said you can see the TX light flickering, looks like the firmware is trying to send something out (the boot message).
Let's try the basics instead:
1. unplug the tinyg USB cable, open the Windows device manager. Write down which devices are listed in the communication ports (COM and LPT) category.
2. plug in tinyg USB and power. Do you hear the Windows-sound for new device connected? Look again in device manager, see if a new COM device is now listed (something like "USB serial device COM8"). If you don't have a new device, check the USB cable, use a different USB port, check and/or update the FTDI device driver (see https://github.com/synthetos/TinyG/wiki/1.02-Connecting-TinyG).
3. start CoolTerm, go to Options -> Serial Port. Make sure it is set to the COM port you found in step 2. Write down the old settings and change to baudrate=115200, data bits=8, parity=none, stop bits=1, flow control = nothing. Hit connect. If there is an error message, recheck step 2.
4. press Enter a few times, try to enter "?" or $h a few times. If you get an answer, go to step 99. If you just get some garbled strings, the baudrate is probably incorrect (retry with different values).
If you get no answer: Hold down the Enter key and look at the TX and RX Leds on the tinyg board. If they are flickering, there is basic communication. Try different baudrates. If none of them is flickering, recheck step 2.
5. If you can't get it to work, you will have to send it back to check/repair/update. Even without a working firmware, you should get some flickering on the RX Led.
99. After you have communication:
Enter $fb to get the firmware build.
Enter $ex to see what kind of flow control is enabled in tinyg. Change the settings in CoolTerm to match that, and save your CoolTerm settings.
You may want to return the board to Synthetos anyway so they can install the bootloader on it.
I hope that helps.