It can be tricky to get a Mac to display video of the early boot process to an external display. It drives me crazy every time when I encounter a broken screen with one of the Macs I support for my organization. It is hard enough to do when booting a full version of macOS, but even more difficult when trying to boot the installer or access any other special startup screens & modes.
Do you still have your original SSD you can boot from? If so, then boot from it and enable display mirroring within the OS. Even this can be tricky. Then try booting from the installer (online or USB won't make any difference since both will behave the same...the only difference is being able to start booting from the installer). Hopefully when you swap SSDs again, you can still see video on the external screen while booting the installer since it may retain the display mirroring option in the NVRAM (aka PRAM).
You can also try a PRAM Reset (hold it for at least three chimes).
You can also try closing the lid of the laptop immediately after you hear the startup chime....use the external keyboard to immediately hold down the necessary keys to boot the installer. You can also try booting the installer with the lid open, then once it gets to the utilities screen closing the lid to see if it will trigger Clamshell mode. You can also try to toggle display mirroring once the installer boots by pressing Command + F1 (needs to be booted first).
Or maybe the Thunderbolt port on the laptop is dirty and not allowing a good connection. Did you ever use the external display before the built-in display broke?
I know some external displays can have idiosyncrasies. Try using the external display's controls to toggle the video inputs. Turn the external display off & back on to see if it will communicate with the laptop to display something while the laptop is in clamshell mode.
It usually just takes persistence and the correct timing.
