I don’t see built-in speakers as an option in the Output window, what to do?

Please check if you are able to see the option in safe mode.

https://support.apple.com/en-us/HT201262

Note: Safe mode prevents your Mac from loading certain software as it starts up, including login items, system extensions not required by macOS, and fonts not installed by macOS. It also does a basic check of your startup disk, similar to using First Aid in Disk Utility. And it deletes some system caches, including font caches and the kernel cache, which are automatically created again as needed.