You need to have a properly partitioned and formatted drive installed internally in the computer in order for the macOS firmware installer to work. I verified this last year with Mojave so I'm assuming the same holds true for the Catalina installer.
My guess is the original SSD is still Ok, but Catalina modified the partition and volume structure which makes it unrecognizable by older versions of macOS (perhaps even Catalina itself). macOS can have trouble recognizing drives if the partition and volumes are in an unknown state.
You can try using Disk Utility to erase the physical SSD to a GUID partition with APFS file system. Recent versions of macOS Disk Utility hide the physical drive so you must first click on "View" within Disk Utility and select "Show all devices". You should now be able to see the physical SSD drive in the left pane of Disk Utility which should appear as the brand and model number of the drive.
If you cannot see the physical drive in Disk Utility, then run the following command in the Terminal app to see if the SSD shows up:
diskutil list
Depending how you boot the computer you may see just a couple of items or you may see 20 items. Depending on the version of macOS you may be able to narrow the list by filtering the list using the following command:
diskutil list internal
IIRC, usually the internal drive will show up as "disk0", but this may not always be the case. If the SSD shows up, then you can zero out the beginning of the SSD to destroy the current partition table by using the following command making sure to substitute the correct drive identifier in place of "diskN".
dd if=/dev/zero of=/dev/diskN bs=100m count=10
If the SSD is listed as "disk0" (that is a zero), then the command would appear like this:
dd if=/dev/zero of=/dev/disk0 bs=100m count=10
If the Terminal show a "$" on the command prompt instead of a "#", then you need to add "sudo " without the quotes before the "dd" in the previous commands.
You should now be able to easily erase the drive using Disk Utility.
If you still have problems, let us know as I have another way to check if the SSD is good or if it has failed.
Good luck.