USB C to thunderbolt SSD's don't show up, but USB C to USB A Hub plugged into the thunderbolt port do show up.

Your title says that the SSD is a Thunderbolt SSD, but your post says that the SSD works when plugged into a hub using "a type C to USB-A cable." That makes me think that the SSD is a USB-C SSD, not a Thunderbolt SSD. USB-A doesn't support Thunderbolt protocol, let alone Thunderbolt speeds.

My next thought is that the USB-C to USB-C cable is suspect. USB-C cables aren't all created equal. There are some, like Thunderbolt-4-certified cables, that can handle almost anything you can throw at them. At the other end, you might find a USB-C cable that only supports charging, or only supports charging + USB 2.0 data rates.

Another possibility is that you have an Apple-Silicon-based MacBook Air, and an Apple security feature is kicking in.

Allow accessories to connect to Mac - Apple Support

"If you have a Mac laptop with Apple silicon, you have to approve new USB or Thunderbolt devices and SD cards that you connect to your Mac."

This affects directly-connected accessories and hubs/docks, but not accessories plugged into hubs:

https://www.macrumors.com/2022/06/06/usb-c-accessories-user-permission-macos-ventura/

So my guess is that either (1) you've got to give permission for the drive to work when directly connected, or (2) your USB-C cable isn't up to the job, and needs to be replaced by a better one.