How did you set up the new SSD? It sounds like you used Disk Utility to "Restore" the original drive to the new SSD, but Disk Utility doesn't work well to "clone" macOS boot drives.
Your two options are to install macOS to the new SSD (installed internally), then use TM to restore from the backup. You can either boot the online macOS installer from Internet Recovery Mode (Command + R for booting the online installer for the OS currently installed, or Command + Option + R to access the Big Sur installer). However, Internet Recovery Mode will sometimes only boot the online installer for the OS which originally shipped with the laptop from the factory which won't work with the Aura NVMe SSD. You can create a bootable macOS USB installer using the instructions in this Apple article:
Create a bootable installer for macOS - Apple Support
Or use the third party app Carbon Copy Cloner to clone the macOS boot drive. You will first need to physically install the Aura SSD and boot from the original Apple SSD externally so you can run CCC (make sure to properly erase the new physical SSD as GUID partition and APFS (top option), you may need to click "View" within Disk Utility and select "Show All Devices" so the physical drives appear on the left pane of Disk Utility). This option requires using an OWC Envoy Pro Enclosure with the original Apple OEM SSD (this enclosure is not compatible with the Aura SSD....AFAIK there are no enclosures or adapters for the Aura SSD).