The question is if there is an alternative router to the Time Capsule that you'd recommend that I can plug in the G-drive?
The Synology router has a USB 3.0 port that will support Time Machine backups. Some of the upper end Asus models do as well. I'm leaning toward the Synology product myself in the next 6-12 months. I think LaPastenague may be using a Synology router now, or he has at least tested one carefully.
this is a self-powering 1 TB SSD so no external power.
If you want to continue to try to get this to work on the Time Capsule, try a powered USB hub. Compatibility wise, I would look for a powered USB 2.0 hub rather than a 3.0 hub.
The USB 3.0 port on your Mac is a much better match for the G-Drive, and it has a lot more "juice" than the USB port on the Time Capsule, which is likely one of the reasons that things are not working correctly on the Time Capsule.
LaPastenague suggested that it was the right format but maybe this needs to be reformatted to work with an older Airport?
If the drive works when it is connected directly to your Mac, then the drive is formatted correctly. So, that is not the issue.
The main likely issues now remain as:
1) The old USB 2.0 port design compatibility question with newer USB 3.0 drives
2) Insufficient power at the USB port on the Time Capsule
Even if you can get things to work with the Time Capsule, I would consider this a very short term answer. A new router is the way to go if you are looking for a reliable longer term solution.